two-dimensional spectroscopy in solid, liquid,...

1

TWO-DIMENSIONAL SPECTROSCOPY

IN SOLID LIQUID AND SURFACE

Doctoral Thesis

YUKI NAGATA

Department of Chemistry Graduate School of Science

Kyoto University

2

TABLE OF CONTENTS

ACKNOWLEDGEMENTS 4

OVERVIEWS 5

1-1 Two-dimensional Raman Spectroscopy 5

1-2 Two-dimensional Surface Spectroscopy 7

1-3 Reference10

TWO-DIMENSIONAL RAMAN SPECTRA OF ATOMIC LIQUIDS AND SOLIDS 13

2-1 Introduction13

2-2 Computational Details 15

2-3 Fifth-order Symmetric and Antisymmetric Integrated Response Functions 17

2-4 Symmetric and Antisymmetric Integrated Response Function for Normal Mode Analysis21

2-5 Temperature Dependence23

2-6 Conclusion 29

2-7 Reference30

ANALYZING ATOMIC LIQUIDS AND SOLIDS TWO-DIMENSIONAL RAMAN SPECTRA IN

FREQUENCY DOMAIN32

3-1 Introduction32

3-2 Frequency-domain Signals of Fifth-order Raman Spectroscopy 33

3-3 Analyzing the Two-dimensional Signals of Lennard-Jones Systems38

3-4 Analyzing the Two-dimensional Signals of Soft Core Systems41

3-5 Conclusion 45

3-6 Appendix Derivation of the Anharmonic Contribution in Morse Potential System45

3-7 Reference48

TWO-DIMENSIONAL INFRARED SURFACE SPECTROSCOPY FOR CO ON CU(100)

DETECTION OF INTERMOLECULAR COUPLING OF ADSORBATES 50

4-1 Introduction50

4-2 Molecular dynamics simulation 53

4-3 Linear Response Function55

3

4-4 Two-dimensional Response Function 56

4-5 Concluding Remarks 62

4-6 Appendix Summary of RESPA 63

4-7 Reference65

CONCLUSION 68

5-1 Quantitative Analyses Beyond Qualitative 68

5-2 Application of the multi-dimensional spectroscopy to surface spectroscopy 69

5-3 Reference71

4

ACKNOWLEDGEMENTS

I wish to express my thanks to Prof Yoshitaka Tanimura in Kyoto University Prof Shinji

Saito in Institute of Molecular Science (IMS) and Prof Shaul Mukamel in University of

California Irvine for overall discussions I also wish to thank Prof Mark Maroncilli in

Wisconsin University for useful comment Dr Hisao Nakamura in the University of Tokyo for

discussion on surface theory Dr Kazuya Watanabe Prof Yoshiyasu Matsumoto in IMS and

Dr Takeshi Yamada in Osaka University on surface experiments I am grateful to Mr Yohichi

Suzuki Mr Taisuke Hasegawa in Kyoto University Mr Hitoshi Maruyama in Ricoh Inc and

Dr Hiroto Kuninaka in Chuo University for much support Some parts of the numerical

calculation were performed at the Computer Center in the IMS

5

Chapter 1

OVERVIEWS

1-1 Two-dimensional Raman Spectroscopy Since the advent of laser technology optical spectroscopy has moved into a revolutionary

new era Development of nonlinear ultrafast spectroscopy in particular has provided exciting

new opportunities for researches in a lot of unexplored areas of science and technology At the

same time the development enables us to perform the new spectroscopic methods One of

these spectroscopic methods is two-dimensional (2D) Raman spectroscopy which is

analogous to 2D nuclear magnetic resonance[1] [2]

2D Raman spectroscopy has been expected to be useful for extracting homogeneous

broadening from inhomogeneous broadening[2] [3] [4] [5] [6] [7] The experiments of 2D

Raman spectroscopy however were very difficult because their signals were very weak and

removing the cascading signal was difficult[8] The successful demonstration of 2D Raman

spectroscopy were done by using CS2 [9] [10] [11] and benzene [12] both of which have

strong polarizabilities

In spite of these experimental difficulties theoretical researches have been continued

actively because 2D Raman spectroscopy has possibility to reveal the underlying dynamics

which cannot be observed in linear Raman spectroscopy Okumura and Tanimura showed the

2D Raman signal was originated from the nonlinear coordinate dependence of polarizability

and anharmonicity of potentials[13] Saito and Ohmine performed the normal mode

calculation of the 2D Raman signals which included only the nonlinear polarizability

contributions However the anharmonicity of potential contribution was not negligible Ma

and Stratt calculated the 2D Raman signal for Xe from the molecular dynamics (MD)

simulations [15] based on a stability matrix formalism[1] [16] In this method the fluctuation

of polarizability of a system which is related to the 2D Raman response functions via the

fluctuation dissipation theorem was calculated[17] The calculated 2D signals from MD

6

simulation with the stability matrix formalism [18] reasonably agreed with experimantal

results for liquid CS2 system[9] On the other hand 2D Raman signals were calculated from a

non-equilibrium MD simulation by Jansen et al in which the laser field was explicitly

radiated into the system[19] [20] [21] In many cases this non-equilibrium MD method is

computationally less expensive than the MD simulation based on the stability matrix

formalism Then Hasegawa and Tanimura developed a less computationally expensive MD

simulation algorism by incorporating the non-equilibrium method into the equilibrium MD

method with the fluctuation-dissipation theorem and calculated 2D signals for various

liquids[22]

The approximations to calculate the 2D Raman signal have been also developed Keyes et

al used the Langevin equation formalism[23] [24] DeVane et al developed the time

correlation function theory and applied it to the CS2 [25] and atomic liquids [26] to calculate

the 2D Raman signals Denny and Reichman used the microscopic molecular hydrodynamics

theory for the 2D Raman spectrum of Xe in which the multi-point correlation function was

factorized into the products of density correlation functions[27] [28] The comparison of the

results using the microscopic molecular hydrodynamics theory with MD simulation results

[15] showed the complete lack of an echo in the hydrodynamic theory These incomplete

descriptions of the 2D Raman signals with these approximated methods may indicate the

importance of the multi-point correlation functions and the stability matrix

Saito and Ohmine investigated the effects of anharmonic dynamics [29] [30] and stability

matrix[29] According to the Okumura and Tanimurarsquos prescription [13] they focused on 1t

and 21 tt = axes for the detection of anharmonic dynamics and stability matrix contributions

respectively To compare the 2D Raman signal quantitatively Nagata and Tanimura projected

2D Raman signals to 1D plots without loss of information on the stability matrix using the

symmetric and anti-symmetric integrated response functions[31] They found the dynamics of

soft core particles were different for in the liquid phase and in the solid phase even in the

ultrafast time scales which could not be observed in linear Raman spectra The dynamics was

composed of the two modes one was a localized mode and the other a delocalized mode[32]

These modes were confirmed in the frequency domain 2D Raman maps more clearly than 1D

plots[33] Moreover they showed that the phase change of the 2D Raman signals was

7

originated from the nonlinear polarizability contribution rather than anharmonicity

contribution [33] which indicated that the nonlinear polarizability was sensitive to the

structural change Temperature dependence of the 2D Raman spectra was also investigated in

the framework of the time correlation function theory[34]

As we overview these theoretical works on the 2D Raman spectroscopy it has potentials to

investigate the intermolecular interaction and related dynamics in condensed phases through

nonlinear polarizability and anharmonic dynamics Such measurements are however difficult

and therefore new innovations on experimental techniques are called for Alternatively some

theoretical research themes should be focused on as a target of the 2D Raman spectroscopy

Ionic liquids [35] or liquid mixtures such as water and acetonitrile [36] [37] [38] are

interesting systems to carry out the 2D Raman measurements Although the sensitivity of the

2D Raman spectroscopy to the dynamics between solid and liquid phases was investigated in

the present thesis the sensitivity to the dynamics near the critical temperature is also

interesting issue I believe further researches in such problems make the future of the 2D

Raman spectroscopy brilliant

1-2 Two-dimensional Surface Spectroscopy The subject of CO bonding and intermolecular dynamics on metal surfaces has attracted

considerable attention with much of the information about the nature of the bond being

inferred from measurements of the intramolecular vibrational mode

As the first stage of theoretical studies the mechanism of the fast relaxation of the CO

stretching mode on metal surfaces was investigated By using the Anderson model Persson

and Persson showed an electron-hole (e-h) pair creation mechanism contributed to the fast

relaxation compared with the phonon effects or the dipole-dipole interactions[39] e-h pair

creation was explained as the action of the induced dipole moments on metal surface caused

by the large dynamic dipole moments of CO The vibrational lifetime estimated was estimated

as ( )13plusmn ps which accords with the experimental results[40] Head-Gordon and Tully

explained the e-h pair creation as a non-adiabatic process Based on the Fermirsquos golden rule

they obtained the following expression for the vibrational lifetime

8

( ) ( )[ ]GGPPTr FF+minus= εεπ

τh

1 (11)

where ( )minusFP ε and G represented the local density of state below Fermi energy Fε and

scattering matrix respectively Equation (11) is analogous to Fisher and Lee formula [42]

[43] and Seideman and Miller formula[44] This analogy indicates CO adsorbed on metal

surfaces works as the scattering source for free electrons of metal surfaces because the CO

creates discrete energy levels against the continuous energy levels of metal surfaces[45]

Moreover Head-Gordon and Tully evaluated the vibrational lifetime from the ab initio

calculation by replacing the ( )minusFP ε and ( )+FP ε with the highest occupied molecular orbital

(HOMO) and lowest unoccupied molecular orbital (LUMO)[46]

Following these studies MD simulation in the dilute CO limit was carried out by Tully et

al[47] They treated e-h pair creation as a stochastic process The residence and random force

were calculated from the first principle calculation through Eq (11) and the second

fluctuation-dissipation theorem respectively They concluded that the relaxation of the CO

stretching and CO frustrated rotational modes arose from e-h pair creation while those of the

C-Cu stretching and CO frustrated translational modes were affected by both Cu lattice

vibration and e-h pair creation As the following researches MD simulations in the medium

coverage were performed to investigate the latest coupling between adsorbed molecules on

surfaces[48] [49]

Experimentalists have also made continuous effort to reveal the underlying dynamics

caused by the interactions between adsorbed molecules Interactions between adsorbed

molecules play an important role on the geometry and lateral hopping of CO on Pd(100)

surface Moreover particular attention has been paid to study the overtone and combination

bands of CO on Ru(001) Broadening of the overtone frequency of the CO stretch rapidly

grows with increasing temperature due to thermally activated decay of two-photon bound

state into single phonon states[50] Large anharmonicities due to lateral coupling lead to the

formation of localized two-phonon bound states besides a continuum of delocalized

two-phonon states[51] Investigating the CO-CO interaction is crucial to reveal the underlying

mechanism of surface dynamics

Here it is hopeful to apply the 2D techniques to surface spectroscopy as the 2D infrared

9

(IR) spectroscopy has succeeded the direct observation of coupling between different modes

In fact such new attempts in surface spectroscopies have been actively done for several

decades Guyot-Sionnest combined the photon echo techniques with sum-frequency

generation (SFG) [52] [53] to separate the homogeneous linewidth from an inhomogeneous

broadening for Si-H vibrations on Si(111) surface[54] M Bonn et al determined the

intermolecular coupling strength of dipole-coupled CO molecules on Ru(001) using

infrared-infrared-visible (IR-IR-VIS) SFG[55] However IR-IR-VIS SFG is a third-order

nonlinear optical process and is not surface-specific N Belabas and M Joffre demonstrated a

visible-infrared 2D spectroscopy in AgGaS2[56] C Voelkmann et al combined four-wave

mixing with the second-harmonic generation to monitor the temporal evolution of

photoexcited one- or two- photon coherence[57] These experiments have two time intervals

between pulses to obtain the information which cannot be accessed by spectroscopies with

one time interval

Nagata and Tanimura investigated the second-order 2D IR surface spectroscopy and

revealed the CO latest coupling is dramatically changed by the frustrated rotational mode

using MD simulation[58] The response function in this spectroscopy has the following form

( ) ( ) ( )[ ] ( )[ ] 0 12

2

12)2( ttittR cbaabc minus⎟

⎠⎞

⎜⎝⎛= micromicromicroh

(12)

where ( )tamicro is the dipole moment at time t in a direction[1] [59] [60] 1t and 2t are

the time intervals between first and second interactions with IR pulses and between the

second interaction and the detection of the signal respectively Since the fifth-order Raman

response function

( ) ( ) ( )[ ] ( )[ ]12

2

12)5( 0 ttittR efcdababcdef minusΠΠΠ⎟

⎠⎞

⎜⎝⎛=h

(13)

has the same form as the response function of surface 2D spectroscopy similar techniques and

analyses developed in 2D Raman spectroscopy are expected to be useful in 2D IR surface

spectroscopy[2] [13] [15] [16] [18] [19] [22] [31] [33]

The potentials of these new spectroscopies are not limited to the metal surface but extends

from the gas-liquid interfaces[61] [62] and liquid-solid interfaces [63] to biological system

such as membrane systems Since the signals from the surfaces however are weaker than

10

those form bulk system multi-dimensional spectroscopies on surfaces may have much

difficulties by means of experiments Nevertheless the author believes the importance and

necessity of 2D IR surface spectroscopy will not be faded out

1-3 Reference [1] S Mukamel Principles of Nonlinear Optical Spectroscopy (Oxford University Press

1995)

[2] Y Tanimura and S Mukamel J Chem Phys 99 9496 (1993)

[3] K Tominaga and K Yoshihara Phys Rev Lett 74 3061 (1995)

[4] K Tominaga and K Yoshihara J Chem Phys 104 4419 (1996)

[5] T Steffen and K Duppen Phys Rev Lett 76 1224 (1996)

[6] T Steffen and K Duppen J Chem Phys 106 3854 (1997)

[7] A Tokmakoff and G R Fleming J Chem Phys 106 2569 (1997)

[8] D A Blank L J Kaufman and G R Fleming J Chem Phys 111 3105 (1999)

[9] L J Kaufman J Heo L D Ziegler and G R Fleming Phys Rev Lett 88 207402

(2002)

[10] V Astinov K J Kubarych C J Milne and R J D Miller Chem Phys Lett 327

34 (2000)

[11] K J Kubarych C L Milne S Lin V Astinov and R J D Miller J Chem Phys

116 2016 (2002)

[12] C L Milne Y L Li T l C Jansen L Huang and R J D Miller J Phys Chem B

110 19867 (2006)

[13] K Okumura and Y Tanimura J Chem Phys106 1687 (1997)

[14] S Saito and I Ohmine J Chem Phys 108 240 (1998)

[15] A Ma and R M Stratt Phys Rev Lett 85 1004 (2000)

[16] S Mukamel V Khidekel and V Chernyak Phys Rev E 53 R1 (1996)

[17] R Kubo M Toda and N Hashitsume Statistical Physics 2 (Springer 1991)

[18] S Saito and I Ohmine Phys Rev Lett 88 207401 (2002)

[19] T l C Jansen J G Snijders and K Duppen J Chem Phys 113 307 (2000)

11


[21] T l C Jansen J G Snijders and K Duppen Phys Rev B 67 134206 (2003)

[22] T Hasegawa and Y Tanimura J Chem Phys 125 074512 (2006)

[23] T Keyes and J T Fourkas J Chem Phys 112 287 (2000)

[24] J Kim and T Keyes Phys Rev E 65 061102 (2002)

[25] R DeVane C Ridley B Space and T Keyes J Chem Phys 119 6073 (2003)


[27] R A Denny and D R Reichman Phys Rev E 63 065101 (2001)

[28] R A Denny and D R Reichman J Chem Phys 116 1987 (2002)



[31] Y Nagata and Y Tanimura J Chem Phys 124 024508 (2006)

[32] H R Schober and B B Laird Phys Rev B 44 6746 (1991)

[33] Y Nagata T Hasegawa and Y Tanimura J Chem Phys 124 194504 (2006)

[34] R DeVane C Kasprzyk B Space and T Keyes J Chem Phys 123 194507

(2005)

[35] H Shirota A M Funston J F Wishart and E W Castner Jr J Chem Phys 122

184512 (2005)

[36] H Kano and H Hamaguchi J Chem Phys 118 4556 (2003)

[37] S Shigeto H Kano and H Hamaguchi J Chem Phys 112 064504 (2005)

[38] E D Elola and B M Ladanyi J Chem Phys 125 184506 (2006)

[39] B N J Persson and M Persson Solid State Communications 36 175 (1980)

[40] R Ryberg Phys Rev B 32 2671 (1985)

[41] M H- Gordon and J C Tully J Chem Phys 96 3939 (1992)

[42] D S Fisher and P A Lee Phys Rev B 23 6851 (1981)

[43] S Datta Electronic Transport in Mesoscopic Systems (Cambridge University Press

1995)

[44] T Seideman and W H Miller in J Chem Phys 96 4412 (1992)

[45] H Nakamura Private Communication

[46] M H- Gordon and J C Tully Phys Rev B 46 1853 (1992)

12

[47] J C Tully M Gomez and M H- Gordon J Vac Sci Technol A 11 1914 (1993)

[48] C Springer and M H- Gordon Chem Phys 205 73 (1996)

[49] M -C Marinica H Le Rouzo and G Raseev Surf Sci 542 1 (2003)

[50] P Jakob Phys Rev Lett 77 4229 (1996)

[51] P Jakob and B N J Persson J Chem Phys 109 8641 (1998)

[52] X D Zhu and Y R Shen Appl Phys B 50 535 (1990)

[53] Y R Shen Principles of Nonlinear Optics (Willy 1984)

[54] P Guyot-Sionnest Phys Rev Lett 66 1489 (1991)

[55] M Bonn C Hess J H Miners T F Heinz H J Bakker and M Cho Phys Rev

Lett 86 1566 (2001)

[56] N Belabas and M Joffre Opt Lett 27 2043 (2002)

[57] C Voelkmann M Reichelt T Meier S W Koch and U Hofer Phys Rev Lett 92

127405 (2004)

[58] Y Nagata Y Tanimura and S Mukamel J Chem Phys submitted (2007)

[59] M Cho J Chem Phys 112 9978 (2000)

[60] R Venkatramani and S Mukamel J Phys Chem B 109 8132 (2005)

[61] P B Miranda and Y R Shen J Phys Chem B 103 3292 (1999)

[62] A Morita and J T Hynes Chem Phys 258 371 (2000)

[63] S Fujiyoshi T Ishibashi and H Onishi J Phys Chem B 108 10636 (2004)

13

Chapter 2

TWO-DIMENSIONAL RAMAN SPECTRA OF

ATOMIC LIQUIDS AND SOLIDS

2-1 Introduction One important aspect of molecular vibrational spectroscopies is the ability to monitor

ultrafast molecular dynamics controlled by complex inter- and intra-molecular interactions[1]

Vibrational relaxations are in principle dependent upon the configurations of atoms therefore

we may expect that the information about the local environments of molecules can be

obtained by analyzing changes in spectra as functions of conditions such as phase density

and temperature However conventional linear spectroscopy does not reveal such changes

because the large broadening caused by damping and inhomogenity makes spectral peaks

featureless To conquer these difficulties 2D vibrational spectroscopies such as fifth-order

Raman spectroscopy [2] [3] [4] and third-order IR spectroscopy [5] have been proposed In

fifth-order Raman spectroscopy a system is perturbed by two pairs of Raman pulses separated

by period t1 and then probed after another period t2 whereas in the third-order IR

spectroscopy a system is perturbed by three IR pulses separated by periods t1 and t2 and then

probed after another period t3 These 2D spectroscopies enable us to evaluate the detailed

interrogation of the interactions and configurations between molecules because the

contributions to the signals from harmonic vibrational motions vanish in multi-dimensional

spectroscopies due to the fact that the Gaussian-integral involves in three-point correlation

function ie ( ) ( )[ ] ( )[ ]0 121 ΠΠ+Π ttt for the fifth-order Raman processes and the

destructive interferences between vibrational excitations in four-point correlation function ie

( ) ( )[ ] ( )[ ] ( )[ ]0 121321 micromicromicromicro tttttt +++ for the third-order IR spectroscopy where ( )tΠ and

( )tmicro are a polarizability and a dipole moment at time t A very large body of theoretical

works on 2D Raman and IR spectroscopies have been devoted to the study of

14

inhomogenity[1] anharmonicity[6] [7] [8] and so on MD simulations have explored

ranging from liquids [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] to more complex

molecules such as peptides[19] [20] [21] [22]

2D Raman spectroscopies are advantageous in studying molecular dynamics in condensed

phases because Raman pulses can create instantaneous vibrational excitations on the

molecular system and their coherence can be detected by spectroscopic means[23] [24] [25]

[26] The sensitivities of the 2D spectra to anharmonicity of the potential and nonlinear

dependence of polarizability on nuclear coordinate have been theoretically demonstrated by

use of the simple models and have been clarified to some extent with the help of quantum

Liouville pathway treatments[27] [28] [29] But most of these studies have not yet provided

helpful pictures of the processes giving rise to particular spectral features in both

experimental data and MD simulations Moreover there has been little guidance from theory

on how to distinguish motions of solids and liquids based on these spectroscopies

In this study we performed MD simulations to investigate how the fifth-order response

functions measured in 2D Raman spectroscopy depended on temperature and thermodynamic

state We have chosen the soft-core model for these simulations because its scaling property

allows us to discuss the phase transition as a function of temperature[30] [31] [32] Note that

the soft-core potential has been used to model metallic glasses with soft modes and has been

one of the widely acknowledged models explaining so-called ldquoboson peakrdquo[33] We explored

the use of the symmetric and anti-symmetric expressions of the integrated 2D Raman

response functions to clarify the interpretation of the spectroscopy data These functions were

originally introduced for an easy check of the simulation results[34] We then projected the

2D profiles onto two kinds of 1D plots one is the classical fifth-order response functions on

the t1=t2 axis [16] and the other is the anti-symmetric integrated response function on the t1=t2

axis Because these functions can be constructed from experimental data as well as simulation

results they will be valuable for analyzing the effects of nonlinear dynamics for instance as

result from anharmonicity of the potentials upon the fifth-order signals The results of our

MD simulations indicate that 2D Raman spectroscopy can detect the change in the character

of molecular motions in different phases in a way that cannot be observed in third-order

Raman spectroscopy On the other hand the profile of the 2D Raman signal is not sensitive to

15

the temperature changes as long as the system is in the same phase Moreover when the

symmetric integrated response function which has the form of a simple three-body

correlation function is compared with the anti-symmetric one with the form of a three-body

correlation function including the stability matrix it is realized what a critical role the

stability matrix plays in extracting dynamical information from the fifth-order response

function

In Sec 2-2 we explain our model and simulation procedures In Sec 2-3 we introduce the

symmetric and anti-symmetric integrated response functions and in Sec 2-4 we analyze the

calculated signal with these functions In Sec 2-5 we discuss the temperature and phase

effects on the signals Finally Sec 2-6 is devoted to concluding remarks

2-2 Computational Details We perform microcanonical MD simulations with a periodic boundary condition on a

system with 108 spheres interacting via a soft potential [33]

( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)

where ε and σ are the potential parameters and the constant A and B are chosen to

connect the force and potential smoothly at the cutoff r0 Thus A and B are given by

( )10023 rσεsdot and ( )6025 rσεsdotminus respectively The molecular system is controlled by laser

pulses The optical response of the system is then described by a correlation function of the

polarizability The total polarizability is treated using a dipole-induced dipole (DID) model

which can be expressed as [35] [36]

( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)

where mα is the molecular polarizability of atom m The second term is important

because it possesses the information on the configuration of the surrounding particles The

third- and fifth-order response functions ( )1)3( tRabcd and ( )12

)5( ttRabcdef which are

associated with the 1D and 2D Raman spectroscopy respectively are given as follows[1]

16

( ) ( ) ( )[ ]011)3(

cdababcd titR ΠΠ=h

(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)

where ( )tabΠ is the ab tensor element of the polarizability at time t [ ] ABBABA ˆˆˆˆˆˆ minusequiv is the

quantum commutator and L is an ensemble average over an equilibrium initial

distribution In the classical limit the operators E and F commute with ˆe Hλ and we have

[10]

( ) ( ) ( )011)3(

cdababcd ttR ΠΠminus= ampβ (25)

( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)

where BPL denotes the Poisson bracket producing stability matrices ( ) ( )02 lk qtp partpart

representing how large the deviation of the momentum of atom k at time t2 is caused by the

slight displacement of atom l at time 0 [37] The significance of the fifth-order measurement

arises from the correlation function including the stability matrix since the stability matrix

carries the information on the interference of the particle trajectories that cannot be obtained

from the third-order measurement

There are two approaches to evaluate these signals using MD simulations the equilibrium

approach [11] [15] [16] and the no-equilibrium approach[13] [14] [16] Here we adopt the

former because it best reveals the importance of the stability matrix to determine the nature of

the fifth-order Raman response In the stability matrix approach we first carry out the

equilibrium simulation then evaluate the response function using the trajectories of particles

obtained from the simulations

To perform the simulation we set 01=ε 01=σ and 01=m without loss of

generality where m is the particle mass Because of the scaling property of the soft-core

potential a temperature times the Boltzmann constant kT is chosen as a parameter with fixed

density 01=ρ We use a fourth-order symplectic integrator method with a time step of 001

The particles form an face-centered cubic (fcc) lattice for 190lekT and they behave like

17

liquid for kTle190 There is also a body-centered cubic (bcc) phase around 019kT asymp but

the region of this phase is relatively narrow[31] [32] Here we calculate the third- and

fifth-order response functions in the fcc solid and liquid phases The MD simulations start

from an fcc configuration with double the targeted temperature and cools the system by

velocity rescaling at the rate of 0002 for each time step After the kinetic energy of the system

reaches kT=0001 we heat the system up to the targeted temperature In both cases we

sample over 60000 trajectories by preparing different initial configurations to calculate the

response functions and we use more than 3500 time steps to stabilize the kinetic energy As

the polarizability is independent of momentum only ( ) ( )12 tqtp rrpartpart of the stability matrix

element is calculated which decreases to one-fourth the computational cost of evaluating the

full stability matrices and reduces the memory requirements for sampling trajectories

2-3 Fifth-order Symmetric and Antisymmetric Integrated Response Functions As has been shown in many previous studies the 2D profiles of the fifth-order Raman

signals are very sensitive to the anharmonicity of the molecular dynamics and nonlinear

dependence of the polarizability But interpreting the 2D profiles in terms of the underlying

dynamics is difficult due to the complex connection between the dynamics and spectroscopy

Since the most significant and interesting contribution to the fifth-order signals comes from

the term with the stability matrix[16] it is valuable and versatile if we can separate it from the

others For this purpose we utilize the symmetric integrated response function introduced by

Cao et al[34] and the anti-symmetric integrated response function

To simplify the following explanation we choose z direction for all tensors and afterward

we omit the tensor notation To make the symmetric form with respect to t1 and t2 from the

fifth-order response function we integrate Eq (26) with respect to t1

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1

ΠΠΠminusminusΠΠΠ=

minusΠΠΠ+ΠΠΠminusminusΠΠΠminus=

equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　

　　　　　　 ampampβ

β

(27)

18

Then by using the relations

( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2

)5( ΠΠΠminus=ΠΠΠ=intinfin

BPttdtttR ampββ

(28)

and

( ) ( ) ( ) ( ) ( ) ( ) 1221 00 BPBP tttt minusΠΠΠ=minusΠΠΠ (29)

we have the symmetric integrated response functions as

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS

minusΠΠΠminusΠΠΠ+minusΠΠΠ=

minusequivampampamp βββ　　　　

(210)

Here we also introduce its counterpart the anti-symmetric integrated response function

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212

ΠΠΠminusminusΠΠΠ+

minusΠΠΠminusminusΠΠΠ=

+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)

Although no special attention has previously been paid to the anti-symmetric integrated

response function we find that it contains the key to analyze liquid dynamics because it

isolates the contribution of the stability matrix from that of the simple three-point correlation

function In the following we demonstrate this point by using the present model in the solid

and liquids phases If the above expressions are described with the normal mode the last two

terms in Eqs (210) and (211) negate with the terms such as ΠΠprimeΠprime (see Eq (23)) Thus

the leading terms of the symmetric and anti-symmetric integrated response functions both

involve ΠprimeΠprimeΠ primeprime

As is evident from the definition these functions satisfy the relations

( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)

Moreover the functions are orthogonal to one another

( ) ( ) 0 121221 =sdotintint ttSttAdtdt (214)

which indicates the information of the symmetric integrated response functions is independent

of that of the anti-symmetric ones

The symmetric integrated response function S(t2t1) is useful for self-consistent checks of

numerical simulations[34] because it does not involve the stability matrix and its numerical

19

calculation is 1N2 times faster than that of the full fifth-order response function On the

contrary calculation of the anti-symmetric integrated response function A(t2t1) requires the

same cost as calculation of the fifth-order response function But it contains the important

information on coherent molecular motions described by the stability matrix It is of

importance to notice that although the present analysis is based on the MD simulation one

can also construct the symmetric and anti-symmetric integrated response functions from

experimental data By utilizing these functions one can quantitatively discuss the effects of

nonlinear dynamics for instance from anharmonicity

To illustrate the nature of these functions we calculate them from the soft-core model in

the solid and liquid phases In Fig 21 we plot the fifth-order response function ( )12)5( ttR

the symmetric and anti-symmetric integrated response functions S(t2t1) and A(t2t1) and the

derivatives of S(t2t1) and A(t2t1) with respect to t1

( ) ( ) 11212)5( tttSttRS partpartequiv (215)

and

( ) ( ) 11212)5( tttAttRA partpartequiv (216)

for kT=0155 in the solid phase To compare the contributions of the symmetric and

anti-symmetric components of the fifth-order response function in Fig 22 we plot the

diagonal slices ( )ttR )5( ( ) ( )21

112)5(

tttS tttSttR==

partpartequiv and ( ) ( )21

112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12

)5( ttRS ( )12)5( ttRA Notice that ( ) ( ) ( )12

)5(12

)5(12

)5( ttRttRttR AS +=

The position of the peak appearing in ( )12)5( ttR is not necessarily located on the t1=t2

axis In fact we can estimate the peak position in Fig 21(a) is ( ) ( )640770 12 =tt by using

a parabolic interpolation Analyzing the fifth-order signals is difficult because 2D profiles of

the signals are usually featureless and the locations of their peaks do not necessarily

correspond to specific physical processes Although the profiles can be changed with the

physical conditions it is very hard to choose portions of the 2D signals to make comparisons

Since the anti-symmetric integrated response function always exhibits a symmetric peak

along t1=t2 axis and is sensitive to the physical conditions due to the contribution from the

stability matrix term it is more instructive to analyze the signals using it instead of the

fifth-order response function In the following we discuss the parameter dependences of the

2D Raman signals with the help of the anti-symmetric integrated response function

20

Figure 21 (a) ( )12

)5( ttR (b) S(t2t1) (c) A(t2t1) (d) ( )12)5( ttRS and (e) ( )12

)5( ttRA at kT=0155 in

the solid phase The peak position at ( ) ( )640770 12 =tt is denoted by a double circle in Fig 1(a)

21

Figure 22 Diagonal slices (t=t1=t2) from the signals depicted in Fig 21 ( )ttR )5( (solid curve)

( )ttRS )5( (dotted) and ( )ttRA )5( (dashed)

2-4 Symmetric and Antisymmetric Integrated Response Function for Normal Mode Analysis

In a normal mode analysis molecular motions are represented by an ensemble of

oscillatory motions along the nuclear coordinates[1] [38] [39] [40] This analysis is often

useful since it gives us access to the microscopic dynamics related to a specific vibrational

frequency[41] Since the fifth-order experiments have the ability to measure anharmonic

vibrational motions through the stability matrix term it is interesting to observe how the

symmetric and anti-symmetric integrated response functions contribute to the signals in the

normal mode analysis

When the polarizability is given in powers of a molecular coordinate as

( ) ( ) ( ) ( )( ) ( ) ( )( ) L+minusΠ primeprime+minusΠprime+Π=Π 202100 qtqqtqt (217)

the third- and fifth-order response functions ( )( )13 tR and ( ) ( )12

5 ttR can be expressed as

( ) ( ) ( ) ( ) ( )ωω

ωωρ 111

3 sin tdtCtR 　intpropprimeprimeΠprimeΠprime= (218)

( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin

ωωωωωωρ ttttd

ttCtCtCttR++

prop

+primeprime+primeprimeprimeprimeΠprimeΠprimeΠ primeprime=

int 　　　　　 (219)

22

where ( ) ( ) ωωttC sin=primeprime We then obtain the following results on t=t1=t2

( ) 0 =ttS (220)

( ) ( ) ( )3

3sinωωωωρ tdttA 　intprop (221)

Since S(tt) gives zero we substitute ( )ttRS )5( and ( )ttRA )5( for S(tt) and A(tt)

( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin

ωωωωωρ ttdttRS (222)

( ) ( ) ( ) ( )intprop 2

2)5( cossin

ωωωωωρ ttdttRA (223)

( ) ( )ttR 5 ( )ttRS )5( and ( )(5) AR t t can be expressed in terms of ( ) ( )tR 3 as

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)

( ) ( ) ( ) ( )( )int minuspropt

A tRtRdtttR0

3)3()5( 339 (226)

Thus by comparing ( )ttRS )5( and ( )(5) AR t t calculated from the fifth-order response

functions with ( )( ) ( )( ) ( )( )( )int ++minust

tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can

easily estimate the contribution from the anharmonic dynamical terms resulting from the

stability matrix and the validity of the normal mode expressions In Fig 23 we compare ( ) ( )ttR 5 ( )ttRS )5( and ( )(5) AR t t with their expressions in terms of ( ) ( )tR 3 under the

normal mode assumption Figure 23(a) clearly shows us that the normal mode expressions

fail to predict the fifth-order response functions even in the short time region However if

we focus on Fig 23(b) we see that the normal mode expressions for the symmetric

integrated response functions are valid in the region 100let The normal mode analysis can

describe the accurate dynamics in the short time which is consistent with the results of the

symmetric integrated response functions One of the reasons for the large deviation at

100get can be deduced from Brownian motion theory When the function ( )tC primeprime is

expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus

minus=primeprime minus (227)

in the Brownian oscillator model the quantities calculated from the fifth-order response

function decay more rapidly than the predictions using the third-order response function On

the other hand in Fig 23(c) ( )(5) AR t t deviates from ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 even for

100let which indicates that the anharmonic contribution from the stability matrix

( ) ( )12 )0()( tqptq minuspartpartΠprimeΠprimeΠprime amp plays an important part in the anti-symmetric integrated

response function Moreover the deviation of the fifth-order response function from the

normal mode expression shown in Fig 23(a) is mainly caused by the anti-symmetric part

2-5 Temperature Dependence We discuss the response functions in the solid and liquid phases at different temperatures

In Fig 24(a) we depict the third-order response functions ( ) ( )tR 3 for kT=0215 and 020 in

the liquid phases and for kT=0155 and 014 in the solid phases[42]

The profiles of the third-order signals in Fig 24 are similar but their decay times are

slightly different for different temperature[43] The fifth-order response functions ( ) ( )125 ttR

are shown in Fig 25 To illustrate the difference between solid and liquid phases we also plot ( )( )ttR 5 for different temperatures as 1D maps in Fig 26 There are differences between

Figs 25(a) (b) in the liquid phases and (c) (d) in the solid phases for 201 get and 202 get

Figure 26 clearly shows that the phase change leads to a change of the spectral decay rates

especially for 150get while the temperature change has a limited effect as long as the

system is in the same phase

24

Figure 23 (a) ( ) ( ) 5 ttR (solid line) and ( ) ( ) ( ) ( ) ( ) ( )( )int minus+t

tRtRtRdt0

333 22436 (dotted line) (b)

( ) )5( ttRS (solid line) and ( ) ( ) ( ) ( ) ( ) ( )( )int ++minust

tRtRtRdt0

333 2433 (dotted line) and (c) ( )(5) AR t t

(solid line) and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 (dotted line) at kT=0155 in the solid phase In each figure

the intensities of the solid lines are normalized at the first peak and the dotted lines are scaled so that their

derivatives are same as those of solid lines near t=0

25

Figure 24 The third-order response functions ( ) ( )tR 3 are depicted for kT=0215 (solid line) 020

(dotted line) in liquid phases and kT=0155 (dashed line) 014 (dot-dash line) in solid phases The

intensities are normalized

Figure 25 The classical fifth-order response functions ( )12

)5( ttR for the cases (a) kT=0215 (b)

kT=020 in the liquid phase and (c) kT=0155 (d) kT=014 in the solid phase The intensity of each plot is

normalized at the first peak

26

Figure 26 The diagonal elements ( )ttR )5( are illustrated for kT=0215 (solid line) 020 (dotted line) in

the liquid phase and kT=0155 (dashed line) 014 (dot-dash line) in the solid phase The intensity of each

plot is normalized at the first peak

Although we can observe differences in the fifth-order response functions between different

phases it is more convenient to employ the symmetric integrated response function S(t2t1)

and the anti-symmetric one A(t2t1) so as to see the effects of the stability matrices In Figs 27

and 8 we depict these functions The diagonal elements of the anti-symmetric integrated

response functions A(tt) are also plotted in Fig 29 While we observe similar tendencies

among the results of S(t2t1) in Figs 27(a)-(d) we see the clear difference among the results

of A(t2t1) in the liquid and solid phases in Figs 28(a)-(d) in 201 get and 202 get and Fig

29 in 150get Although we cannot do longer simulations due to the limitations to our

computational power we may expect the bigger and clearer differences in the decay rates in

the anti-symmetric response functions appear in the longer time region The sensitivity of

A(t2t1) and insensitivity of S(t2t1) to the phase change result from the stability matrix which

reveals deviations from harmonic motion as the interference between the trajectories The

anti-symmetric integrated response functions which can be obtained from experimental data

as well as simulation results by following the procedure explained in Sec 2-3 are more

valuable than the symmetric ones because the former allow us to estimate the contribution

from the stability matrix directly

27

Figure 27 The symmetric integrated response functions ( )12 ttS are plotted as 2D contour maps for the

cases (a) at kT=0215 (b) at kT=020 in liquid phase and (c) at kT=0155 (d) at kT=014 in solid phase The

intensity of each figure is normalized at the first peak

28

Figure 28 The differences of the anti-symmetric integrated response functions ( )12 ttA are plotted as

two-dimensional contour maps for the cases (a) at kT=0215 (b) at kT=020 in liquid phase and (c) at

kT=0155 (d) at kT=014 in solid phase The intensity of each plot is normalized at the first peak

Figure 29 The diagonal elements ( )ttA are illustrated for kT=0215 (solid line) 020 (dotted line) in

liquid phase and kT=0155 (dashed line) 014 (dot-dash line) in solid phase The intensity of each plot is


29

2-6 Conclusion Using the MD simulation we calculated the third- and fifth-order Raman signals of atomic

solids and liquids described by the soft-core potential To analyze these signals quantitatively

and to reveal the effect of the stability matrix we have decomposed the classical fifth-order

response function into the symmetric and anti-symmetric integrated response functions the

symmetric one has the form of the simple three-body correlation function while the

anti-symmetric one also includes contributions from the stability matrix The latter can not be

expressed accurately in terms of normal modes expressions even in the short time region

where the symmetric integrated response function can be This fact indicates that the

anharmonic contributions which are missing in the normal mode expressions give rise to an

important effect on the fifth-order signals

It is shown that the change between liquid and solid phases causes a minor effect on the

third-order signals while it causes dramatic differences in the fifth-order signals The

anti-symmetric integrated response functions show prominent changes as a result of the phase

transition whereas the change in the symmetric one is modest The difference between the

symmetric and anti-symmetric integrated response functions results from the sensitivity of the

stability matrix to the nonlinear dynamics because the stability matrix reveals the deviation

from the harmonic motion as the interference between the trajectories This result suggests the

advantage of using the anti-symmetric integrated response functions rather than the fifth-order

response functions for analysis Such features are however unable to be revealed within the

frame work of the normal mode analysis since it neglects the effects from the anharmonicity

described by the stability matrix

Since we can always construct the symmetric and anti-symmetric integrated response

functions not only from the simulation results but also from the experimental data these

functions will be valuable and versatile tools for analyzing the effects of nonlinear dynamics

upon the fifth-order signals

30

2-7 Reference [1] Y Tanimura and S Mukamel J Chem Phys 99 9496 (1993)


34 (2000)


116 2016 (2002)


(2002)

[5] P Hamm M Lim and R M Hochstrasser J Phys Chem B 102 6123 (1998)


[7] Y Tanimura Chem Phys 233 217 (1998)

[8] R B Williams and R F Loring J Chem Phys 113 1932 (2000)

[9] S Ryu and R M Stratt J Phys Chem B 108 6782 (2004)


[11] A Ma and RM Stratt Phys Rev Lett 85 1004 (2000)

[12] A Ma and RM Stratt J Chem Phys 116 4972 (2002)





[17] A Piryatinski and J L Skinner J Phys Chem B 106 8055 (2002)

[18] C P Lawrence and J L Skinner J Chem Phys 119 3840 (2003)

[19] K Kwac H Lee and M Cho J Chem Phys 120 1477 (2004)

[20] K A Merchant W G Noid D E Thompson R Akiyama R F Loring and M D

Fayer J Phys Chem B 107 4 (2003)

[21] A M Moran S -M Park and S Mukamel J Chem Phys 118 9971 (2003)

[22] T Hayashi and S Mukamel J Phys Chem A 107 9113 (2003)


[24] M Cho Two-dimensional vibrational spectroscopy in Advances in Multi-photon

Process and Spectroscopy ed SH Lin AA Villaeys and Y Fujimura (World

31

Scientific Singapore 1999) vol 12 p 229

[25] S Mukamel Annu Rev Phys Chem 51 691 (2000)

[26] J T Fourkas Adv Chem Phys 117 235 (2001)

[27] T Steffen J T Fourkas and K Duppen J Chem Phys 105 7364 (1996)

[28] K Okumura and Y Tanimura J Phys Chem A 107 8092 (2003)

[29] W G Noid and R F Loring J Chem Phys 121 7057 (2004)

[30] W G Hoover S G Gray and K W Johnson J Chem Phys 55 1128 (1971)

[31] W G Hoover D A Young and R Grover J Chem Phys 56 2207 (1972)

[32] B B Laird and A D J Haymet Mol Phys 75 71 (1992)


[34] J Cao S Yang and J Wu J Chem Phys 116 3760 (2002)

[35] P A Madden and D J Tildesley Mol Phys 55 969 (1985)

[36] B M Ladanyi Chem Phys Lett 121 351 (1985)


[38] M Buchner BM Ladanyi and RM Stratt J Chem Phys 97 8522 (1992)

[39] RM Stratt and MH Cho J Chem Phys 100 6700 (1994)２

[40] RM Stratt Acc Chem Res 28 201 (1995)

[41] M Cho GR Fleming S Saito I Ohmine and RM Stratt J Chem Phys 100

6672 (1994)

[42] Since we consider a small system of 108 particles in the present simulations there

are large fluctuations of temperature These fluctuations cause the difficulty in

carrying out the simulation near the phase transition temperature kT=019 because

during the simulation the molecular states may take both solid and liquid phases due

to the fluctuation of the temperature Therefore we avoid the regions at near kT=019

in our simulations

[43] K Okumura B Bagchi and Y Tanimura Bull Chem Soc Jpn 73 873 (2000)

32

Chapter 3

ANALYZING ATOMIC LIQUIDS AND SOLIDS

TWO-DIMENSIONAL RAMAN SPECTRA

IN FREQUENCY DOMAIN

3-1 Introduction Nonlinear optical interactions between the laser and molecular system provide valuable

and versatile spectroscopic information to understand the dynamics of the system as well as

its environment For molecules in a condensed phase fifth-order Raman and third-order IR

spectroscopies allow us to capture greater details in molecular dynamics and structure than

third-order Raman and first-order IR spectroscopies[1] The utility and possibility of these

spectroscopies have been demonstrated by various approaches including theoretical

analyses[2] [3] MD simulations[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

[18] [19] [20] ab initio calculations[21] [22] [23] [24] and a variety of experiments[25] [26]

[27] [28] [29] [30] [31] [32] [33] Although the potential of multi-dimensional vibrational

spectroscopies is now well recognized our comprehension of 2D contour maps has not been

achieved completely In a fifth-order Raman case this might be attributed to the lack of theory

explaining an overall profile of signal The previous chapter showed how one could utilize the

symmetric and anti-symmetric integrated response functions [34] to characterize the role of

stability matrix in 1D plots These functions cannot separate the contribution of nonlinear

polarizability from that of anharmonicity of potentials but can isolate the contribution of the

stability matrix from that of the simple three-point correlation function The role of stability

matrix is important in terms of equilibrium simulation whereas the comparison of the term of

nonlinear polarizability with that of anharmonicity is essential with respect to the normal

mode (NM) analysis

33

This chapter presents a practical method to evaluate the relative contributions of nonlinear

polarizability and anharmonicity of potentials from the experimental and simulation data

These contributions can be estimated from analytical theories [3] or MD simulations [6][13]

by turning on and off the terms responsible for the contributions however it has not been

possible to analyze directly the experimental data especially for multi-mode systems due to

the complication of the 2D maps Our approach can extract quantitative information about the

ratio between the nonlinear polarizability and anharmonicity of potentials in the molecular

system To evaluate this ratio we have derived the analytical expression of the fifth-order

Raman signal for an anharmonic potential system based on a perturbative calculation of a

Morse oscillator system[3] [13] By carrying out the double Fourier transformation of the

2D time domain Raman signal we have obtained the frequency domain expressions of the

fifth-order Raman signal The analysis of spectral volumes gives access to the ratio between

nonlinear polarizability and anharmonicity which reveals the change of molecular

mechanisms between solid and liquid phases [7]

In Sec3-2 we have derived the expression of fifth-order response functions in the

frequency domain The derivation of the analytical expression for anharmonicity term is

explained in Appendix In Sec3-3 we have applied our method to simple liquids described by

a Lennard-Jones (LJ) potential and showed that we can clarify each vibrational mode

uniquely by using 2D frequency domain maps The NM expressions using the calculated ratio

between nonlinear polarizability and anharmonicity are compared with MD simulation

results[13] In Sec3-4 we have further applied our method to the soft core potential systems

to investigate a role of nonlinear polarizability and anharmonicity which was not clarified in

our previous study[7] Section 3-5 is devoted to the concluding remarks

3-2 Frequency-domain Signals of Fifth-order Raman Spectroscopy The fifth-order Raman response function is defined by a three-body correlation function of

polarizability In this section we demonstrate how one can obtain the information of the

nonlinear polarizability and anharmonicity from the experimental and simulation data by

using the analytical expression of the response function

34

We assume the polarizability with ab tensor ( )tabΠ is expanded in terms of a single

molecular coordinate ( )tq as

( ) ( ) ( ) L+Π primeprime+Πprime=ΠminusΠ 2

2tqtqt abababab

λ (31)

where λ is a perturbation index which is set to unity after completion of the perturbation

expansion The molecular coordinate is treated as a harmonic motion ( )tqH plus a

perturbative anharmonic motion ( )tqA

( ) ( ) ( )tqtqtq AH λ+= (32)

The fifth-order response function ( ) ( )125 ttRabcdef is expressed as

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125

125 ttRttRttRttR ANabcdefNLabcdefLTabcdefabcdef ++= λ (33)

where

( ) ( ) ( ) ( )12125

01 tqtMkT

ttR HHefcdabLTabcdef minusΠprimeΠprimeΠprime= amp (34)

and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef

minusminusΠ primeprimeΠprimeΠprime+

minusΠprimeΠ primeprimeΠprime+

minusΠprimeΠprimeΠ primeprime=

amp

amp

amp

　　　　　　　

　　　　　　　 (31)

are the contributions from linear and nonlinear polarizability and

( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM

kTttR AHefcdabHAefcdabANabcdef

minusΠprimeΠprimeΠprime+minusΠprimeΠprimeΠprime= ampamp (36)

is that from anharmonicity of potentials Here ( ) ( ) ( )121212 ttMttMttM AH λ+= is the

stability matrix for harmonic and anharmonic parts of trajectories and kT represents the

temperature multiplied by Boltzmann constant

In the Brownian oscillator (BO) model the linear polarizability term vanishes and the

nonlinear polarizability term is reduced to [2] [3] [4]

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef

+primeprimeprimeprimeΠprimeΠprimeΠ primeprime+

primeprimeprimeprimeΠprimeΠ primeprimeΠprimeprop

int

int

ωωρω

ωωρω

　　　　　　　

(37)

35

where ( ) ( ) tettC γω minus=primeprime sin and the decay rate γ is assumed to be independent of ω in

order to simplify the following discussion Hereafter we adopt the BO model with an

exponential decay and do not take a bi-exponential decay The expressions of decay functions

do not make much difference on the following procedure and discussion about the ratio

between nonlinear polarizability and anharmonicity of potentials

Since the fifth-order signal calculated from the BO approach does not involve the

anharmonic contribution it disagrees with the direct evaluation of the response function by

means of MD simulations In order to improve the BO approach and evaluate ( ) ( )125

ttR ANabcdef

we analytically calculate ( )tqH and ( )tqA for a Morse potential

( ) ( )qq eeDqV ββ minusminus minus= 22 (38)

from the perturbative expansion approach where β and D are potential parameters We

then derived the anharmonicity part of the fifth-order response function as

( ) ( ) ( )int ⎟

⎠⎞

⎜⎝⎛ +primeprime⎟

⎠⎞

⎜⎝⎛primeprimeΠprimeΠprimeΠprimeprop

224 2

123

2125

ttCtCdttR efcdabANabcdef ω

ωρωβ (39)

An explicit derivation of Eq (39) is given in Appendix where we compare Eq (39) with the

anharmonic part of the fifth-order response function for the anharmonicity additive potential

and show the accordance of Eq (39) with the expressions obtained by Okumura and

Tanimura and by Ma and Stratt when 33 6 βDg minus= [3] [13]

The anharmonic term (39) together with the nonlinear term (37) casts the total fifth-order

response function into the form

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞

⎜⎝⎛primeprimeΠprimeΠprimeΠprimeminus



224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)

If ca = and db = we can recast Eq (310) as

( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞

⎜⎝⎛primeprimesdot+

+primeprime+primeprimeprimeprimeprop int

224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36

where the intensities of the nonlinear polarizability and anharmonicity are now denoted by

( )ωNLabcdefk and ( )ωANabcdefk respectively As is illustrated below ( )ωNLabcdefk and

( )ωANabcdefk can be evaluated quantitatively from the experimental and simulation data with

use of the double Fourier transformation of the response function defined by

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)

The substitution of Eq (311) into Eq (312) leads to the expression without tensor elements

as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω

ωωωωωωωωωωωωρωωω

~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++

++prop int　　　　　　　

　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω

γωωω+minus+minus

=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~

γωωγωωωωωωγ

ωωω+minus+minus

minusminusminussdot=NLR (316)

( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus

minusminusminus=ANR (318)

and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus

minusminus+minussdot=ANR (319)

If we set ( ) ( )aωωδωρ minus= the signal consists of two peaks centered at aωωω == 21 and

aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~

γωωγωωγωωω

γωωγωωγωωωωω

aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)

In a real system the spectral distribution ( )ωρ may not be a delta-function and the terms

such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω

ωωωωωωρωd (321)

can also make a contribution to the spectra Therefore the profiles and positions of peaks are

slightly displaced from those predicted by Eq (320) In the case that ( )ωρ has a Gaussian

like profile however the contributions of Eq (321) to the spectral volume centered at

aωω asymp is negligible because we have

( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp

+minus+minusminusminus

int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω

ωω γωωγωωωωωω

ωωρωωω a

a

a

a

ddd (322)

where ω∆ represents the spectral width If a frequency-resolved spectrum obtained from

either the experiments or simulations is expressed as ( ) ( ) ( ) ( ) ( ) ( )aaobsaaobsobs RRR ωωωωωωωωωω asympasymp+asympasymp= 12

512

512

5 2~~~ (323)

the contributions from the nonlinear polarizability and anharmonicity for each molecular

motion can be evaluated through the following equation

( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)

We now consider the method to separate the peak near aωωω == 21 from other peaks for

example near aωωω == 221 If ( ) ( ) ( ) ( )215

125 ~~ ωωωω obsobs RR minus is calculated Eqs (315) and

(318) vanish because of the symmetry with respect to 1ω and 2ω for any ( )ωρ

Moreover by using ( ) ( ) ( ) ( )215

125 ~~ ωωωω obsobs RR minus we can remove the cross peaks which often

overlap with the peak at 221 ωω = in a multimode system The volume of each peak can be

measured accurately as illustrated in Secs 3-3 and 3-4 Hereafter we discuss zzzzabcd =

and zzzzzzabcdef = tensors in the response functions and drop the suffixes for simplicity

38

3-3 Analyzing the Two-dimensional Signals of Lennard-Jones Systems

Using 2D frequency domain maps we can also clarify the peaks more clearly than 1D

frequency domain plots In this section we apply our method to the fifth-order 2D Raman

signals of the LJ potential system which were well investigated by Ma and Stratt [12][13] to

evaluate the consistency of the ratio NLAN kk At the same time we demonstrate a way to

judge whether the spectral peak is composed of a single mode or multi-modes

We calculate the fifth-order Raman signals in the LJ potential system by the equilibrium

and non-equilibrium hybrid method[11] Then we transformed the signals from time domain

to frequency domain with a Welch window function MD simulations are carried out with 108

LJ atoms The same potential polarizability size of the simulation box and time step used in

these simulations are used as those of Ma and Strattrsquos simulation[12] [13] The forces by LJ

potential and DID interaction are smoothly cut off at the half length of the simulation box

with the switching function In the non-equilibrium calculations the system is irradiated by

laser pulse pairs with the strength of 50 VA The third- and fifth-order response functions are

calculated by averaging over 4000000 configurations

The 1D frequency domain plot of the third-order response function ( ) ( )ω3~R is shown in

Fig 31 We found that the signal may be fit by either one Lorentzian peak or three Gaussian

peaks which are depicted in Fig 31(a) and (b) respectively[30] Figures 32(a) and (b) show

the 2D contour plots in frequency domain ( ) ( )125 ~ ωωR and ( ) ( ) ( ) ( )21

512

5 ~~ ωωωω RR minus

respectively These figures indicate that the dynamics in the LJ potential system is governed

by one mode Thus ( ) ( )ω3~R should be fit with a single Gaussian peak as Fig 31(b) and the

frequency for the translational motion is found to be about 19 cm-1

As shown in Fig 32(b) ( ) ( ) ( ) ( )215

125 ~~ ωωωω RR minus allows us to specify the peak near

19221 == ωω cm-1 more easily than ( ) ( )125 ~ ωωR does The calculated ratio of the spectral

volumes ( ) ( ) ( ) ( )215

215 ~2~ ωωωω == RR is 300minus which gives us 04=NLAN kk for the

translational mode In fact if we adapt this ratio and calculate the diagonal element on

21 tt = and 2t axis element on 01 =t of Eq (310) according to the NM expressions using

the third-order response function ( ) ( )tR 3 as

39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞

⎜⎝⎛ minus+prop

t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and

( )( ) ( )( ) ( )( ) ( )( )( )intint +minus+propt

AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)

we can well reconstruct the original fifth-order response function in the short time region as

plotted in Fig 32(c) The large deviations of Eqs (325) and (326) from MD simulations

over 200 fs are attributed to the use of the NM expression instead of BO expression

Moreover the agreement of these results with Fig 7 in Ref [13] strongly supports the ability

to evaluate the ratio between anharmonicity and nonlinearity from the present our method

Figure 31 The third-order response functions ( ) ( )ω3~R are depicted as the solid line (a) The curve

fitted with one Lorenzian peak and (b) the curves fitted with three Gaussian peaks are plotted as the dashed

lines

40

Figure 32 (a) The fifth-order response function ( ) ( )125 ~ ωωR and (b) ( ) ( ) ( ) ( )21

512

5 ~~ ωωωω RR minus for

the LJ system at 220=T K are depicted In Fig (c) The MD results along 21 tt = (solid line) and

01 =t (dotted line) are compared with the signals calculated from Eqs (325) (solid line with circle

markers) and (26) (dotted line with circle markers) respectively

The temperature dependence of the ratio ANNL kk is investigated at 95=T 120 and

160 K for the supercooled liquids and at 220=T and 260 K for the normal liquids under

same density and is shown in Fig 33[35] In Fig 33 this ratio seems almost constant in

both phases which means that fifth-order response function cannot capture the qualitative

change between both liquids Although the mobility is changed in general between both

liquids its effect does not appear in short time scale of the present simulation

41

Figure 33 The ratio of ANNL kk for the LJ potential system is illustrated as the function of temperature

The dashed line is a guide for eyes

3-4 Analyzing the Two-dimensional Signals of Soft Core Systems We investigate the soft-core potential system defined by the potential

( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)

where ε and σ are the potential parameters and the constants A and B are chosen to


( )10023 rσε and ( )6025 rσεsdotminus respectively We carried out the MD simulation in

reduced units A temperature multiplied by Boltzmann constant is changed from solid phase

( 1550140=kT ) to liquid phase ( 2150230200 　=kT ) as a parameter with fixed density

The other conditions on the simulation are the same as the previously study[7] in which we

found that the fifth-order Raman signals are sensitive to the difference of the phases while the

third-order Raman signals are not The method developed in this study sheds light on the

relative intensities of the nonlinear polarizability and anharmonicity on the fifth-order signals

The 1D time domain and frequency domain plots ( ) ( )tR 3 and ( ) ( )ω3~R 2D time domain

and frequency domain maps ( ) ( )125 ttR and ( ) ( )12

5 ~ ωωR and ( ) ( ) ( ) ( )215


at 140=kT are depicted in Figs 34(a) (b) (c) (d) and (e) respectively and at

200=kT are shown in Fig 35(a) (b) (c) (d) and (e) respectively Figure 34(b) exhibits a

42

similar profile to Fig 35(b) and the 1D frequency domain maps in the other temperatures

These signals are also similar to the LJ case depicted in Fig 31 On the other hand we can

clearly see the difference of 2D signals in the soft-core and LJ cases the 2D signals of

soft-core potential system consist of two modes The frequency of two modes are found to be

56=lω and 16=hω respectively The modes of lω and hω are thought to be from

delocalized vibrational motion for each atom and high frequency localized mode

respectively[36]

Figure 34 (a) The third-order response function ( ) ( )ω3~R and the fifth-order response functions (b)

( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512

5 ~~ ωωωω RR minus for the soft core system at 1400=kT are

depicted (d) The time domain 2D maps ( ) ( )125 ttR are shown for comparison (see reference 9) The

curve fitted to the third-order spectrum with two Gaussian peaks is presented as the dotted line All figures

are in reduced units

43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512





The relative intensities of the nonlinear polarizability and anharmonicity for lω and hω

are determined from the spectral volumes of 2D frequency domain maps as summarized in

Table 1 The calculated ratios for each mode are visualized in Fig 36 for ( ) ( )lANlNL kk ωω

( ) ( )lANhAN kk ωω and ( ) ( )hANhAN kk ωω All ratios except for the nonlinear polarizability at

lω show almost proportional to each other at any temperature while the ratios between

( )lNLk ω and the other intensities exhibit a specific phase dependency as shown in Fig 36

Thus we may consider that the nonlinear polarizability of the delocalized mode is

dramatically changed whereas the high frequency localized mode is not changed as long as

44

the fifth-order Raman spectra can capture This consideration indicates that the difference of

the anti-symmetric integrated response functions between solid and liquid phases [7] is

induced by the nonlinear polarizability of lω In addition the anharmonicity makes larger

contribution to the fifth-order signal than the nonlinear polarizability for the localized mode

while makes smaller contribution for delocalized mode as shown in Table 1

Figure 36 The ratios of ( ) ( )lANlNL kk ωω (square) ( ) ( )lANhAN kk ωω (triangle) and

( ) ( )hNLhAN kk ωω (circle) for the soft-core potential system are illustrated as the function of temperature

All lines are guides for eyes

45

3-5 Conclusion In this section we proposed the method to evaluate the contributions from the nonlinear

polarizability and anharmonicity of potentials utilizing the analytical expression of 2D signals

in frequency domain The ratio between two contributions was evaluated from the volumes of

spectral peaks With 2D Raman frequency domain maps we could easily separate

contributions from different vibrational modes which was difficult with 1D approach

Demonstration to apply our method to the simulation results for LJ potential system

indicates that primary contribution of molecular motions to the signal comes from only a

translational mode at 19 cm-1 Moreover we evaluated the ratio between nonlinear

polarizability and anharmonicity as 04=NLAN kk Good agreement is obtained in

comparison of the NM expression of the fifth-order response functions using this ratio with

the data of Ma and Stratt[13] which supports the consistency of our method

We also examined our method to analyze the signals of the soft-core potential system at

various temperatures In our previous study[7] it was found that the 2D time domain Raman

signals exhibited a clear difference between the solid and liquid phases but the origin of this

difference could not be identified The present method shows that the different fifth-order

Raman signals between both phases resulted from the change of the mode of lω through the

nonlinear polarizability At the same time the phase transition was shown to have little effect

on the mode of hω We can conclude from the fifth-order response function that the

dynamical change between the solid and liquid phases is characterized as not high frequency

localized mode but delocalized motion in soft core potential systems Moreover when our

attention is paid to the contributions to the fifth-order signal we can find that the nonlinear

polarizability and anharmonicity of potential are dominant for the localized and delocalized

modes respectively

3-6 Appendix Derivation of the Anharmonic Contribution in Morse Potential System

The contribution of anharmonicity to the fifth-order Raman signal was first evaluated by

Okumura and Tanimura using Feynman rules for anharmonicity additive potential[3] Then

46

by using the adiabatic instantaneous normal mode theory Ma and Stratt derived the same

expression as Okumura and Tanimura[13] Here we show the same expression can be

obtained from the perturbative calculations by assuming a Morse potential system defined by

Eq (38) When the correspondence of the anharmonicity additive potential

( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)

with the Morse potential is taken into consideration we can obtain the relations

microβω D2= and 33 6 βDg minus= by expanding the Morse potential with respect of q We

may evaluate the fifth-order response function from the solution of the Morse potential

because the contributions of the nonlinear polarizability and anharmonicity over 2λ are

negligible at small q

The equation of motion for the Morse potential system can be solved analytically and the

coordinate and momentum are given by

( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)

where micro is the mass of molecule and 1C and 2C are the integral constants When we

expand Tayler series up to the first-order around the bottom of the Morse potential the

position and momentum are written as

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22

12sin11 CtDCCtDCtq βmicro

βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2

12cos12 CtDCCtDCDtp βmicro

βmicro

micro (332)

Using these solutions we can calculate the stability matrix analytically as

47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)

The substitutions of these equations into Eqs (33) (34) (35) and (36) give us the

expression for the anharmonicity of potentials as

( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛minusΠprimeΠprimeΠprime+

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛minusΠprimeΠ primeprimeΠprime+

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛minusΠprimeΠprimeΠ primeprime=

21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)

When the relations microβω D2= and ( ) 221 221 kTqDC ==minus microω are used Eq (338)

becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞

⎜⎝⎛ +ΠprimeΠprimeΠprimeminus

ΠprimeΠ primeprimeΠprime+

+ΠprimeΠprimeΠ primeprimeprop

2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)

The expression for the anharmonicity additive potential is obtained by setting 33 6 βDg minus=

which has the same form as the expressions obtained by Okumura and Tanimura and by Ma

and Stratt

48


1995)


[3] K Okumura and Y Tanimura J Chem Phys 107 2267 (1997)





[8] T l C Jansen JG Snijders and K Duppen J Chem Phys 113 307 (2000)

[9] T l C Jansen J G Snijders and K Duppen ibid 114 10910 (2001)

[10] T l C Jansen K Duppen and J G Snijders Phys Rev B 67 134206 (2003)



[13] A Ma and R M Stratt J Chem Phys 116 4972 (2002)


[15] R L Murry J T Fourkas and T Keyes J Chem Phys 109 7913 (1998)



[18] A Piryatinski C P Lawrence and J L Skinner J Chem Phys 118 9664 (2003)


[20] R Akiyama and R F Loring J Chem Phys 116 4655 (2002)

[21] S Hahn K Park and M Cho J Chem Phys 111 4121(1999)



[24] T Hayashi T l C Jansen W Zhuang and S Mukamel J Phys Chem A 109 64

(2005)




49

334 (2000)


116 2016 (2002)

[29] K J Kubarych C J Milne and R J D Miller Chem Phys Lett 369 635 (2003)



[32] L J Kaufman D A Blank and G R Fleming J Chem Phys 114 2312 (2001)


(2002)


[35] The supercooled liquids at 95=T 120 and 160 K appear as metastable states

The detailed phase diagrams are shown in J ndashP Hansen and L Verlet Phys Rev 184

151 (1969) G A Vliegenthart J F M Lodge and H N W Lekkerkerker Physica

A 263 378 (1999)


50

Chapter 4

TWO-DIMENSIONAL INFRARED

SURFACE SPECTROSCOPY FOR CO ON

CU(100)

DETECTION OF INTERMOLECULAR

COUPLING OF ADSORBATES

4-1 Introduction The vibrational energy dissipation and intermolecular coupling of CO on various metal

surfaces have attracted much attention and is well understood[1] The CO stretching mode

lifetime can be explained by an electron-hole (e-h) pair creation mechanism due to

antibonding 2π orbital of the CO molecule[2] The frustrated CO rotational mode have

very short lifetimes dominated by the same mechanism[3] [4] [5] Chemical shifts for the

frustrated rotational mode are different for Cu(100) and Cu(111) [6] [7] which reflects the

sensitivity of the adsorbate dynamics to the electronic structure of metal surfaces due to e-h

pair creation The stretching and frustrated rotational modes are anharmonically coupled to

each other[8] [9] At the same time the CO frustrated rotational and translational modes are

coupled[10] These intermode couplings govern the dynamics of adsorbates on metal surface

Dipole-dipole interactions play an important role on the geometry and lateral hopping of

CO on Pd(100) and Pd(110)[11] [12] Researchers have tried to control lateral CO

motions[13] [14] [15] The lateral interactions make local ordering of adsorbates and prevent

the formation of long range ordered structure at low coverage[16] Particular attention has

been paid to the overtone and combination bands of CO on Ru(001) Broadening of the CO

51

stretch overtone rapidly grows with temperature due to thermally activated decay of

two-photon bound state into single phonon states[17] Large anharmonicities due to lateral

coupling lead to the formation of localized two-phonon bound states[18] Thus the

investigation of the intermolecular couplings of CO adsorbed on metal surface is crucial for

revealing the underlying mechanism of the surface dynamics

In linear spectroscopy the couplings between intra- and inter-adsorbate interactions are

measurable only in terms of chemical shifts In addition linear spectroscopy is unable to

distinguish homogeneous broadening from inhomogeneous broadening[19] [20] To overcome

these limitations multidimensional spectroscopic techniques have been proposed and carried

out to obtain information on surface dynamics Guyot-Sionnest combined the photon echo

and sum-frequency generation (SFG) techniques [21] [22] to extract the homogeneous

linewidth from the inhomogeneously broadened Si-H vibrations on Si(111) surface[23] Cho

proposed infrared-infrared-visible (IR-IR-VIS) SFG [24] and Bonn et al determined the

intermolecular coupling strength of dipole-coupled CO on Ru(001) using this SFG[25] [26]

However IR-IR-VIS SFG is a third-order nonlinear optical process and is not surface-specific

Belabas and Joffre demonstrated two-dimensional (2D) VIS-IR spectroscopy in AgGaS2[27]

Voelkmann et al combined four-wave mixing with the second-harmonic generation to

monitor the temporal evolution of photoexcited one- or two- photon coherence[28]

In this study we propose to apply 2D IR surface spectroscopy which utilizes the SFG of

two independently tunable IR beams to an admixture of C12O16 and isotope labeled C13O16 on

Cu(100) The pulse sequence and energy diagram of 2D IR surface spectroscopy is presented

in Fig 1 with those of 2D Raman spectroscopy 2D IR surface spectroscopy has two

independent time axes retaining the surface-specificity and is related to the second-order

response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)

where ( )tamicro is the dipole moment at time t in a direction[19] [29] [30] [31] 1t and 2t

are the time intervals between the first and second IR pulses and between the second pulse

and the signal respectively Since the fifth-order Raman response function

52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)

where ( )tabΠ is the polarizability at time t in ab tensor has the same form as the

second-order IR response function Eq(41) similar formulism and simulation strategies

developed in 2D Raman spectroscopy [20] [32] [33] [34] [35] [36] [37] [38] [29] [40] [41]

[42] [43] are applicable to 2D IR surface spectroscopy We calculate 2D IR surface signals by

means of molecular dynamics (MD) simulation based on the stability matrix formalism [33]

[35] [37] [40] to investigate roles of anharmonic coupling[34] In third-order IR spectra 13C

and 18O isotope substitution has been used to change the frequency [44] [45] [46] [47] [48]

[49] and the cross peaks between two amide-Irsquo modes were observed[49] These suggest the

possible detection of isotopic effects of adsorbates in 2D IR surface spectroscopy

The aim of the present article is to explore a possibility of 2D IR surface spectroscopy to

detect intermolecular anharmonic couplings between CO and isotopic CO stretching modes

on Cu(100) at different temperatures In Sec 42 details of the MD simulation are explained

The temperature dependence of the first-order response functions of IR spectroscopy and the

second order response functions of 2D IR surface spectroscopy are discussed in Sec 43 and

Sec 44 respectively Finally concluding remarks are given in Sec 45

Figure 41 Pulse sequences (left) and typical energy diagrams (right) for 2nd-order IR surface

spectroscopy and 5th-order Raman spectroscopy Solid and dotted lines denote the transition

of bra and ket vectors respectively

53

4-2 Molecular dynamics simulation In 2D Raman spectroscopy researchers have calculated the classical limit of the fifth-order

Raman response function

( ) ( ) ( ) ( )1212)5( 01 tt

kTttR efBPcdababcdef minusΠΠΠminus= amp (43)

where BPL k and T denote the Poisson bracket boltzmann constant and temperature

respectively[33] [35] [36] [37] [38] [39] [40] [41] [42] [43] The response function Eq (3)

gives direct information on anharmonic dynamics [39] [43] and coherent dynamics through

the stability matrix[39] [40] Similarly in 2D IR surface spectroscopy it is necessary to

calculate the classical limit of the second-order IR response function

( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

in order to describe the surface dynamics and reveal the dynamical intermode correlation We

calculated the 2D surface IR signals by means of the MD simulation based on the stability

matrix formalism where calculation of stability matrix ( ) ( )02 ji ptq partpart needs the N3

trajectories for one initial configuration where N is the number of particles

The following potentials were employed in our MD simulation the Cu-Cu interaction

potential developed by Wonchoba and Truhlar [50] the Cu-C-O three-body potential and C-O

intramolecular potential by Tully et al [5] and the CO-CO inter-adsorbate potential by van

der Pol et al [51] and Janssen et al[52] The inter-adsorbate potential included Van der Waals

terms and electronic terms representing dipole-dipole dipole-quadrupole and

quadrupole-quadrupole interactions The effects of e-h pair creation was included as

stochastic forces and frictions whose parameters are made by Tully et al[5] Similar MD

simulations for CO on Cu surface have been carried out to investigate the desorption of CO

from Cu(100) enhanced by neighbor CO molecules [53] and adsorption of CO on stepped

surface of Cu(211)[54]

Our simulations were based on classical equations of motion for CO interacting with 4

surface layers with 16 Cu atoms in each layer The bottom layer was rigid and interacted only

54

with the nearest layer Since the CO intramolecular force was much stronger than the other

forces we employed the RESPA algorithm [55] summarized in the Appendix The time

evolution was carried out for 6121 == tt ps with 1=∆T fs and 10=∆t fs where T∆ and

t∆ are the longer and shorter time steps defined in the RESPA 6121 == tt ps is not enough

to observe the linewidth of CO stretching mode in 1D plots or 2D maps but is sufficient for

the dynamical coupling between adsorbates The Langevin equations were solved using

Brunger-Brooks-Karplus algorithm[56] To calculate the optical response the dipole moment

( )tmicro was assumed to be

( ) ( )trt 10 micromicromicro += (45)

where 0micro and 1micro denoted the permanent dipole moments and the first derivative of the

dipole elements as the function of CO distance ( )tr By using Gaussian03 package the

parameters 132600 =micro Debye and 09891 =micro Debye Å -1 were determined from the

optimized structure of free CO and OCCu with fixed 13 Cu atoms respectively[4] The

optimized geometries and dipole moments were calculated at the B3LYP6-31G(2dp) level of

the density functional theory

The dipole moments for these high frequency modes are primarily determined by the

intramolecular interactions Therefore the first derivatives of CO dipole moments with

respect to Cu atoms degrees of freedom are negligible compared with those with respect to C

and O atoms making it unnecessary to calculate the stability matrix of Cu atoms

In 2D Raman spectroscopy of pure liquids non-equilibrium MD (NEMD) simulations are

computationally less expensive than MD simulations based on the stability matrix

formalism[36] [42] This is because for second-order IR or fifth-order Raman response

functions calculation of stability matrix requires 3N trajectories while only 4 trajectories

are needed in NEMD simulations[42] However when the simulations of multidimensional

spectroscopy are directed to specific vibrational modes such as the CO vibrational mode of

amide I in aqueous solution [49] the above approximation requires only calculation of the

stability matrix with respect to C and O atoms composing the CO vibrational mode and the

atoms around them if needed It should be stressed that the above approximation may make

the MD simulations based on the stability matrix formalism computationally less expensive

than the NEMD

55

The 2D signals were calculated by averaging over 1500000 1200000 and 700000

initial configurations for 60K 100K and 150K respectively Numerical convergence in time

domain data was confirmed by checking the anti-symmetric integrated response functions

which characterized the behavior of stability matrix [40]

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000002 ΠΠΠminusminusΠΠΠminusminusΠΠΠ= BPBPBP ttttttA (45)

The anti-symmetric integrated response functions for 60K 100K and 150K are shown in Fig

42

Figure 42 Anti-symmetric integrated response functions of second-order IR response functions

at 60K 100K and 150K Each plot is normalized at the maximum of the data

4-3 Linear Response Function The MD simulations of the linear response functions were carried out for four COs and

isotopic COs on Cu(100) at 60K 100K and 150K The spectra displayed in Fig 43 show the

CO stretching frequencies of CO and isotopic CO were 12179 minuscm and 12123 minuscm at 150K

respectively The corresponding frequencies for one CO and isotopic CO on Cu(100) are 12175 minuscm and 12120 minuscm This shows blue shift due to the electronic interaction in

satisfactory agreement with the experiment[6]

The peak positions are independent of temperature Indeed temperature has influence on the

linewidth in the experiment but the linewidth is not precisely reproduced in our simulation as

56

stated in the previous section Although the dynamics may be activated by increasing

temperature it has little effect on the CO stretching frequency The spectra are not affected by

the dynamics of neighbor adsorbates

Figure 43 Linear spectra at 60K 100K and 150K Each plot

is normalized as the area is same at each temperature

4-4 Two-dimensional Response Function One of the merits of the 2D techniques is their capacity to identify signals corresponding to

specific Liouville space pathways[19] Here we focus only on CO stretching modes The

double-sided Feynman diagrams corresponding to the peaks on the areas of the fundamental

tone area ( )12

111

1 2250205022502050 minusminusminusminus lelelele cmcmcmcm ωω and overtone area

( )12

111

1 4500410022502050 minusminusminusminus lelelele cmcmcmcm ωω are sketched in Fig 44(a) and (b)

respectively[30]

The calculated 2D signals in the fundamental tone area at different temperatures are

depicted in Figs 45 The main peaks are located at

( ) ( ) ( ) ( ) ( )1111111121 21792179212321792179212321232123 minusminusminusminusminusminusminusminus= cmcmcmcmcmcmcmcmωω

Since the times 6121 == tt ps are shorter than the relaxation time of signals we observe

weak sidelobe peaks along 11 21792123 minus= cmω and 1

2 21792123 minus= cmω

The overtone transitions of a single oscillator such as |0002||0000| gtltrarrgtlt make a

larger contribution to the diagonal peaks than the transitions due to the inter-adsorbate

57

couplings such as |0011||0000| gtltrarrgtlt where gtji| represents the combination of

i th excited state of CO and j th excited state of isotopic CO Thus the insensitivity of

diagonal peaks to temperature indicates that the potential anharmonicity with respect to single

CO stretching mode 31

3111 rVg partpartequiv is unchanged The volumes of the cross peaks are at

most about one tenth of those of the diagonal peaks

Figure 44 Double-sided Feynman diagrams of 2D IR surface response function in CO on Cu(100)

corresponding to the signals (a) in the fundamental tone area and (b) in the overtone area

gtg| gt1| e and gt2| e represent ground state one quanta excited states including gt01| and

gt10| and two quanta excited states including gt02| gt11| and gt20| respectively

White and black circles show once and twice interactions between laser field and system

Figure 45 Imaginary parts of 2D signals of the fundamental tone at (a) 60K (b) 100K and (c) 150K

Each graph is normalized at maxima peaks

58

The cross peaks representing anharmonic intermolecular couplings between CO and

isotopic CO change significantly at 60K 100K and 150K The intermolecular couplings are

caused by intermolecular anharmonic potentials composed by different CO stretch modes[24]

At least three primary causes for inter-adsorbate couplings are possible induced dipole

moments on Cu surface (e-h pair creation) the couplings through surface Cu atoms (phonon

effect) and electronic interactions between CO and isotopic CO The first mechanism is

excluded in our simulation because the effect of e-h pair creation is expressed by the frictions

and stochastic forces [5] and cannot produce the correlation between CO and isotopic CO

Therefore we consider the latter two possibilities To examine the second mechanism we

calculated the 2D signal at 150K by not allowing motion of the Cu atoms The results shown

in Fig 46 resemble to the Fig 45(c) which indicates that the phonon effect is not dominant

for inter-adsorbate couplings To evaluate the third mechanism we calculated the 2D map at

150K with the electronic charges on CO to be zero and illustrated it in Fig 47 The cross

peaks at ( ) ( )1121 21792123 minusminus= cmcmωω in Fig 47 are much weaker than those in Fig 45

and the phase of the cross peaks at ( ) ( )1121 21232179 minusminus= cmcmωω also changes between

two figures The inter-adsorbate interactions are primarily governed by electronic interactions

which are about 10 times stronger than Van der Waals interactions Therefore the temperature

dependence of the cross peaks may be attributed to electronic interactions

Since the electric charges are fixed in our simulation the electronic interaction strength

depends on the distance between COs In Fig 48 we show the radial distribution functions

(RDFs) of C-C O-O and Cu-Cu The C-C and O-O RDFs similarly expand with temperature

in contrast to the Cu-Cu RDF indicating that temperature activates the frustrated rotational

mode The nonlinear dipole moments and anharmonicity of potentials give non-vanishing

elements in the second-order IR response function as in the fifth-order Raman response

function[32] The nonlinearity however plays minor role on the second-order response

function compared to the anharmonicity In contrast both give comparable contributions to

the fifth-order Raman response function[38] [29] [41] [30] Based on these arguments we

focused mainly on the electronic interactions through the dynamics of CO and its potential

anharmonicity Our model is simplified to extract the lateral dipole-dipole couplings on

surface the electric charges qminus on adsorbates and q+ on metal surface are set and the

59

Figure 46 Imaginary parts of 2D signals of the Figure 47 Imaginary parts of 2D signlas of

the fundamental tone at 150K with Cu atoms fixed fundamental tone at 150K without electronic

interaction

Figure 48 Radial distribution functions of (a) Cu-Cu

(b) C-C and (c) O-O at 60K (solid) and 150K (dotted)

angles between dipole moments and vertical line 1θ and 2θ are selected as variables (Fig

49) The inter-adsorbate potential of a right side adosorbate is given by

60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)

where R is the distance between surface atoms and ir is the distance between the i th

adsorbate and the corresponding surface atom The strengths of the cross peaks are

proportional to the potential anharmonicity for different normal modes 221

3122 rrVg partpartpartequiv

[24] [32] [34] We plot 122g in Fig 410 as the functions of 1θ and 2θ with AR 63=

Arr 9121 == and Debyeqr 11 = for simplicity It is found that 122g increases

monotonously as 21 θθ minus increases It is found that 122g increases monotonously as 21 θθ minus

increases The 21 θθ minus distribution for all adsorbates is shown in Fig 411 This indicates that

the rotational dynamics of adsorbates becomes active for higher temperature which leads the

wider 21 θθ minus distribution The wider 21 θθ minus distribution increases the anharmonicity of

potentials 122g and changes the phase of the cross peaks followed by Fig 410

Figure 49 The model with two dipole moments A and S denote the adsorbate and surface atoms

Figure 410 Anharmonicity of potential 122g as functions of 1θ and 2θ

61

Figure 411 ( )21cos θθ minus distribution when Cu-Cu distance is less than 38A

The distributions at 60K and 150K denote solid and dotted lines respectively

Figure 412 Imaginary parts of 2D signals of the overtone at (a) 60K (b) 100K and (c) 150K


Finally the calculated 2D signals in the overtone area at different temperatures are shown

in Figs 412 When Figs 45 and 412 are compared we can find that the cross peaks of Fig

411 are much smaller than those of Fig 45 This can be explained by the double-sided

Feynman diagrams in Fig 44 The cross peaks in the fundamental tone area arise from for

example the optical process |0000||0010||0001||0000| gtltrarrgtltrarrgtltrarrgtlt in

Feynman diagram Fig 44(a1) Similarly the cross peaks in the overtone area arise from for

example the process |0000||0011||0001||0000| gtltrarrgtltrarrgtltrarrgtlt in Fig 44(b)

62

Since the one quantum process |0011||0001| gtltrarrgtlt and two quanta processes

|0010||0001| gtltrarrgtlt and |0000||0011| gtltrarrgtlt require the couplings between CO

and isotoped CO and have much less transition dipole moment than the one quantum

processes |0001||0000| gtltrarrgtlt and |0000||0010| gtltrarrgtlt the cross peaks in the

fundamental tone area are bigger than those in the overtone area Moreover the diagonal

peaks in the overtone area have opposite signs as those in the fundamental tone area As was

shown in the fifth-order Raman spectroscopy [41] the peak intensities of the signals in

frequency domain 2D maps are determined by the interplay between the nonlinear coordinate

dependence of dipole moments (NL) and the anharmonicity if potential (AN) NL+AN

contributes to the diagonal peaks in the fundamental tone area while NL-AN contributes to

the diagonal peaks in the overtone area Since the contributions of NL to 2D signals are much

smaller than those of AN the signs of the diagonal peaks in the fundamental tone and

overtone areas become positive and negative corresponding to the contributions AN and ndashAN

respectively

4-5 Concluding Remarks We carried out the MD simulation for CO on Cu(100) to calculate the signals of 2D IR

surface spectroscopy for various temperatures Our MD simulation was based on the stability

matrix formalism and e-h pair creation was included as a stochastic process When

temperatures were set to be 60K 100K and 150K the cross peaks were significantly changed

in the fundamental tone areas of 2D frequency domain maps while the appreciable difference

could not be found in linear response functions Comparison of the signals from the MD

simulations without electronic interactions and with surface Cu atoms fixed indicated that the

electronic interactions were the primary cause of the temperature dependence of the cross

peaks To explore this point we employed the simple model with two dipole moments whose

configurations were characterized by the angles between dipole moments and vertical line 1θ

and 2θ It was found that the increase of 21 θθ minus changed the anharmonicity from negative

to positive signs In fact wider 21 θθ minus distribution with higher temperature was observed in

our simulation The frustrated rotational mode activated by increasing temperature changed

63

the anharmonicity of potentials and consequently the phase of cross peaks

We further compared the signals in the overtone area with in the fundamental tone area

The cross peaks in the overtone area were much smaller than those in the fundamental tone

area This was because the couplings between CO and isotoped CO were required twice in the

optical process corresponding to the signals in the overtone area while there was one

coupling in the optical process in the fundamental tone area Moreover the phases of the

diagonal peaks were changed between in the fundamental tone area and in the overtone area

This was explained from our previous study of 2D Raman spectroscopy the anharmonicity of

potential with respect to the single CO stretching mode gave positive and negative

contribution to the fundamental tone and the overtone respectively

Finally we make the following two points First the e-h pair creation mechanism plays a

major role in surface dynamics such as pure dephasing on metal surface[57] and theoretical

descriptions of e-h pair creation are well tested[58] [59] Although the effect of e-h pair

creation was treated by using the friction and stochastic forces in this study it is challenging

to simulate the influence of e-h pair creation on the inter-adosorbate couplings Second some

studies have shown the capacity of 2D IR spectroscopy to reveal the intermolecular

interactions by probing the intramolecular interactions [60] [61] [62] [63] For example

Zheng et al probed the equilibrium dynamics of phenol complexation to benzene in a

benzene-carbon tetrachloride solvent mixture [60] [61] and Cowan et al investigated into the

loss of memory of persistent correlations in water structure [62] In this study the temperature

dependence of the inter-adsorbate coupling was discussed by observing the cross peaks of the

CO stretch This study demonstrated the same ability of the multidimensional IR

spectroscopies to provide similar information on surfaces as well as in the bulk

4-6 Appendix Summary of RESPA The RESPA was introduced to lower the computation cost of integrating the equations of

motion by separating all forces into short and long range types sF and lF [55] These forces

are integrated with different time steps In one dimensional system the Liouville operator L

can be rewritten as

64

VlVsK iLiLiLiL ++= (47)

where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)

The propagator can then be written ( ) ( ) ( ) ( )[ ] ( ) ( )323222 TOetOeeeee tiLntiLtiLtiLtiLTiL VlKVsKVl ∆+∆+= ∆∆∆∆∆∆ (411)

where nTt ∆=∆

In our system the force by the CO intramolecular potential corresponds to sF while all

other forces are classified into lF The Liouville operator of the kinetic part can be

decomposed into fast and slow motion parts corresponding to sF and lF respectively

KlKsK iLiLiL += (412)

Note that KsL is dependent on the degrees of freedom of C and O atoms while KlL includes

the degrees of freedom of all atoms Since Cu atoms are independent of sF the time

evolution of Cu atoms can be extracted from the iteration with smaller time step Thus we get ( ) ( )[ ] ( ) ( )( ) ( ) ( ) ( )[ ] ( ) ( )323222

322

　　　　 TOeetOeeee

TOeeeeTiLTiLntiLtiLtiLTiL

TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)

Eq (413) avoiding n times time evolution with respect to the position of Cu atoms

compared with the original RESPA (411)

The increase of the atoms independent of the short range forces accelerates the efficiency

of Eq (413) The methodology is useful in the MD simulations ranging from the surface

dynamics to the molecules in the solution in which the solute molecules are treated as rigid

bodies and interacts with the solute molecules We explain this methodology using the

velocity Verlet algorism which is an example of a second order symplectic integrator This

methodology is more important for the higher order symplectic integrator method

65

4-7 Reference [1] J C Tully Annu Rev Phys Chem 51 153 (2000)

[2] B N J Persson and M Persson Solid State Commun 36 175 (1980)




[6] C J Hirschmugel G P Williams F M Hoffmann and Y J Chabal Phys Rev Lett

65 480 (1990)

[7] C J Hirschmugel and G P Williams Phys Rev B 52 14177 (1995)

[8] J P Culver M Li L G Jahn R M Hochsrasser and A G Todh Chem Phys Lett

214 431 (1993)

[9] T A Germer J C Stephenson E J Heilweil and R R Cavanagh J Chem Phys

101 1704 (1994)

[10] A P Graham F Hofmann J P Toennies G P Williams C J Hirschmugl and J

Ellis J Chem Phys 108 7825 (1998)

[11] H Kato M Kawai and J Yoshinobu Phys Rev Lett 82 1899 (1999)

[12] J Yoshinobu N Takagi and M Kawai Phys Rev B 49 16670 (1994)

[13] H J Lee and W Ho Science 286 1719 (1999)

[14] T Komeda Y Kim M Kawai B N J Persson and H Ueba Science 295 2055

(2002)

[15] L Bartels F Wang D Moller E Knoesel and T F Heinz Science 305 648 (2004)

[16] E Borguet and HL Dai J Phys Chem B 109 8509 (2005)



[19] S Mukamel Principles of Nonlinear Optical Spectroscopy (Oxford University Press

New York 1995)


[21] Y R Shen The Principles of Nonlinear Optics (Wiley-Interscience New York

1984)


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[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)

[26] M H Cho C Hess and M Bonn Phys Rev B 65 205423 (2002)



127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)







[36] TlC Jansen JG Snijders and K Duppen J Chem Phys 113 307 (2000)







[43] S Saito and I Ohime J Chem Phys 125 084506 (2006)

[44] S Woutersen and P Hamm J Chem Phys 114 2727 (2001)

[45] J Bredenbeck and P Hamm J Chem Phys 119 1569 (2003)

[46] C Fang J Wang A K Charnley W Barber-Armstrong A B Smith S M Decatur

and R M Hochstrasser Chem Phys Lett 382 586 (2003)

[47] P Mukherjee A T Krummel E C Fulmer I Kass I T Arkin and M T Zanni J

Chem Phys 120 10215 (2004)

[48] C Fang J Wang Y S Kim A K Charnley W Barber-Armstrong A B Smith S

67

M Decatur and R M Hochstrasser J Phys Chem B 108 10415 (2004)

[49] Y S Kim J Wang and R M Hochstrasser J Phys Chem B 109 7511 (2005)

[50] S E Wonchoba and D G Truhlar J Chem Phys 99 9637 (1993)

[51] A van der Pol A van der Avoird and P E S Wormer J Chem Phys 92 7498

(1990)

[52] W B J M Janssen J Michiels and A van der Avoird J Chem Phys 94 8402

(1991)



[55] M Tuckerman B J Berne and G J Martyna J Chem Phys 97 1990 (1992)

[56] A Brunger C L Brooks and M Karplus Chem Phys Lett 105 495 (1984)

[57] K Watanabe N Takagi and Y Matsumoto Phys Rev Lett 92 057401 (2004)

[58] F Weik A Demeijere and E Hasselbrink J Chem Phys 99 682 (1993)

[59] H Nakamura and K Yamashita J Chem Phys 122 194706 (2005)

[60] J R Zheng K Kwak J Asbury Z Chen I R Piletic and M D Fayer Science 309

1338 (2005)

[61] K Kwak C Lee Y Jung J Han K Kwak J R Zheng M D Fayer and M Cho J

Chem Phys 125 244508 (2006)

[62] M L Cowan B D Bruner N Huse J R Dwyer B Chung E T J Nibbering T

Elsaesser and R J D Miller Nature 434 199 (2005)

[63] J D Eaves J J Loparo C J Fecko S T Roberts A Tokmakoff and P L Geissler

Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION

5-1 Quantitative Analyses Beyond Qualitative Over ten years passed since 2D Raman spectroscopy was proposed theoretically [1] and

some experimental successes were reported[2] [3] [4] [5] By using MD simulations the 2D

Raman signals were calculated in Xe [6] [7] CS2 [7] [8] water [7] [9] [10] benzene [11] and

soft core system[12] When these experimental and simulation results were discussed one

usually employed oversimplified models such as the Brownian oscillator model[13] The

discussions and analyses by using these models were qualitative rather than quantitative

Quantitative analyses beyond qualitative shall be required as the next stage of 2D Raman

spectroscopy to insight into the details of molecular dynamics

Without navigation tools for 2D maps it is difficult to compare experimental data or

simulation data with each other For example although we can obtain some information such

as peak positions and direction of ridges from the 2D maps we cannot extract any

information on dynamics as long as the underlying mechanisms of these features are not

revealed Tthe projection of the 2D maps onto the 1D plots without loss of important

information which characterizes the fifth-order Raman response function is one of the

practical ways to analyze the 2D signals How we project the 2D maps onto 1D plots depends

on what we want to investigate In this thesis we showed two examples if we want to know

the role of the stability matrix analyzing the antisymmetric integrated response function is the

best approach[12] If you want to know the ratio between anharmonicity contributions and

nonlinear polarizability which are the main sources of the 2D Raman signal analyzing the 2D

frequency-domain map is the best way[14] Note that the former approach can extract more

dynamical information than the latter

In chapter 2 we applied these methods to the soft core system and found that even in

femtosecond order ultrafast region 2D Raman signals can capture the difference between solid

69

and liquid phases to which the linear Raman spectra are not sensitive This indicates that 2D

Raman spectroscopy has a possibility to investigate the detailed molecular dynamics and

structures near critical points which cannot be observable in linear Raman spectroscopy

The sensitivity of 2D Raman spectroscopy arises from the anharmonicity of potentials and

nonlinear coordinate dependence of polarizability because harmonic dynamics with linear

polarizability has no contribution to the 2D Raman signals[1] [13] In chapter 3 [14] we

derived the expressions of 2D Raman signals for Morse potential in the Brownian oscillator

model using the perturbative solution of equation of motions If the Fourier transformations of

2D Raman signals are carried out the peaks shall be located along 21 ωω = and 221 ωω =

axes in a 2D frequency domain map By evaluating the volume of each peak we can measure

the ratio between the anharmonicity of potentials and nonlinear polarizability contributions

through Eq (324)

When Fourier transformation was performed for the 2D time domain data of soft core

system we found there were two peaks from the localized and delocalized modes By

measuring the spectral volume in the 2D maps in frequency domain only the nonlinearity

contribution of the localized mode showed dramatic change between liquid and solid phases

while other nonlinearity and anharmonicity contributions were continuously changed between

them This result indicates that the nonlinear polarizability is sensitive to the change in

molecular structure and dynamics We found that only localized mode of the soft core

potential system varied between solid and liquid phases

5-2 Application of the multi-dimensional spectroscopy to surface spectroscopy Both from the experimental and theoretical points of view it is natural to develop

multidimensional surface spectroscopy in the following reasons From the experimental point

of view the information of dynamical inter- and intra-adsorbate coupling is important to

reveal the underlying surface dynamics The inter-adsorbate interactions however appear as

chemical shifts in linear spectroscopy which does not represent the intermode couplings

Here multidimensional spectroscopies for bulk system such as 2D IR and Raman

70

spectroscopies have given us the information on direct anharmonic couplings between

different modes and anharmonic dynamics To investigate the details of inter-adsorbate

couplings on surface 2D techniques should be applied to surface spectroscopy From the

theoretical point of view the second-order IR response function has the same expression as

the fifth-order Raman response function which indicates that similar formalisms or

discussions developed in 2D Raman spectroscopy are applicable to 2D IR surface

spectroscopy According to the previous chapters on fifth-order Raman spectroscopy we can

separate homogeneous broadening from inhomogeneous broaded spectra [1] we can obtain

information on the intermode couplings [15] and we can extract the contributions of

nonlinear coordinate dependence of dipole moments and anharmonicity of potentials with

respect to normal mode by using 2D Raman spectroscopy[13] The information which is

expected to be obtained in 2D IR surface spectroscopy is also important for clarifying the

underlying surface dynamics

Indeed the pioneering researches have targeted at higher-order spectroscopies to detect the

details of surface dynamics by some groups as stated in Sec 4-1 but these experiments have

not explored the merits of 2D technique such as the detection of intermode couplings Here I

considered the detection of intermolecular coupling between adsorbates on metal surface by

using 2D IR surface spectroscopy for demonstration[16] I applied it to an admixture of CO

and isotoped CO on Cu(100) by means of MD simulations based on the stability matrix

formalism

The 2D profiles of the signals in frequency domain showed both diagonal and cross peaks

The former peaks mainly arose from the overtones of the CO and isotoped CO while the

latter represented the couplings between those As temperature increased the phases of cross

peaks in a second-order infrared response function changed significantly while those of

diagonal peaks are unchanged We showed that the phase shifts were originated from

anharmonicity of potentials due to the electronic interaction between adsorbates Using a

model with two-dipole moments we found that the frustrated rotational mode activated with

temperature had effects on the anharmonicity

These results indicate that 2D IR surface spectroscopy reveals the anharmonic couplings

between adsorbates and surface atoms or between adsorbates which can not be observed in

71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






2

TABLE OF CONTENTS

ACKNOWLEDGEMENTS 4

OVERVIEWS 5

1-1 Two-dimensional Raman Spectroscopy 5

1-2 Two-dimensional Surface Spectroscopy 7

1-3 Reference10

TWO-DIMENSIONAL RAMAN SPECTRA OF ATOMIC LIQUIDS AND SOLIDS 13

2-1 Introduction13

2-2 Computational Details 15

2-3 Fifth-order Symmetric and Antisymmetric Integrated Response Functions 17

2-4 Symmetric and Antisymmetric Integrated Response Function for Normal Mode Analysis21

2-5 Temperature Dependence23

2-6 Conclusion 29

2-7 Reference30

ANALYZING ATOMIC LIQUIDS AND SOLIDS TWO-DIMENSIONAL RAMAN SPECTRA IN

FREQUENCY DOMAIN32

3-1 Introduction32

3-2 Frequency-domain Signals of Fifth-order Raman Spectroscopy 33

3-3 Analyzing the Two-dimensional Signals of Lennard-Jones Systems38

3-4 Analyzing the Two-dimensional Signals of Soft Core Systems41

3-5 Conclusion 45

3-6 Appendix Derivation of the Anharmonic Contribution in Morse Potential System45

3-7 Reference48

TWO-DIMENSIONAL INFRARED SURFACE SPECTROSCOPY FOR CO ON CU(100)

DETECTION OF INTERMOLECULAR COUPLING OF ADSORBATES 50

4-1 Introduction50

4-2 Molecular dynamics simulation 53

4-3 Linear Response Function55

3




4-7 Reference65

CONCLUSION 68



5-3 Reference71

4

ACKNOWLEDGEMENTS










5

Chapter 1

OVERVIEWS
























6









liquids[22]






















7




























8


τh

1 (11)

















surfaces[48] [49]












9













one time interval




( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(12)




response function

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(13)



spectroscopy[2] [13] [15] [16] [18] [19] [22] [31] [33]




10





1995)









(2002)


34 (2000)


116 2016 (2002)


110 19867 (2006)








11
















(2005)


184512 (2005)









1995)




12










Lett 86 1566 (2001)



127405 (2004)


[59] M Cho J Chem Phys 112 9978 (2000)





13

Chapter 2
























14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






3




4-7 Reference65

CONCLUSION 68



5-3 Reference71

4

ACKNOWLEDGEMENTS










5

Chapter 1

OVERVIEWS
























6









liquids[22]






















7




























8


τh

1 (11)

















surfaces[48] [49]












9













one time interval




( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(12)




response function

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(13)



spectroscopy[2] [13] [15] [16] [18] [19] [22] [31] [33]




10





1995)









(2002)


34 (2000)


116 2016 (2002)


110 19867 (2006)








11
















(2005)


184512 (2005)









1995)




12










Lett 86 1566 (2001)



127405 (2004)


[59] M Cho J Chem Phys 112 9978 (2000)





13

Chapter 2
























14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






4

ACKNOWLEDGEMENTS










5

Chapter 1

OVERVIEWS
























6









liquids[22]






















7




























8


τh

1 (11)

















surfaces[48] [49]












9













one time interval




( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(12)




response function

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(13)



spectroscopy[2] [13] [15] [16] [18] [19] [22] [31] [33]




10





1995)









(2002)


34 (2000)


116 2016 (2002)


110 19867 (2006)








11
















(2005)


184512 (2005)









1995)




12










Lett 86 1566 (2001)



127405 (2004)


[59] M Cho J Chem Phys 112 9978 (2000)





13

Chapter 2
























14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






5

Chapter 1

OVERVIEWS
























6









liquids[22]






















7




























8


τh

1 (11)

















surfaces[48] [49]












9













one time interval




( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(12)




response function

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(13)



spectroscopy[2] [13] [15] [16] [18] [19] [22] [31] [33]




10





1995)









(2002)


34 (2000)


116 2016 (2002)


110 19867 (2006)








11
















(2005)


184512 (2005)









1995)




12










Lett 86 1566 (2001)



127405 (2004)


[59] M Cho J Chem Phys 112 9978 (2000)





13

Chapter 2
























14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






6









liquids[22]






















7




























8


τh

1 (11)

















surfaces[48] [49]












9













one time interval




( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(12)




response function

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(13)



spectroscopy[2] [13] [15] [16] [18] [19] [22] [31] [33]




10





1995)









(2002)


34 (2000)


116 2016 (2002)


110 19867 (2006)








11
















(2005)


184512 (2005)









1995)




12










Lett 86 1566 (2001)



127405 (2004)


[59] M Cho J Chem Phys 112 9978 (2000)





13

Chapter 2
























14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






7




























8


τh

1 (11)

















surfaces[48] [49]












9













one time interval




( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(12)




response function

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(13)



spectroscopy[2] [13] [15] [16] [18] [19] [22] [31] [33]




10





1995)









(2002)


34 (2000)


116 2016 (2002)


110 19867 (2006)








11
















(2005)


184512 (2005)









1995)




12










Lett 86 1566 (2001)



127405 (2004)


[59] M Cho J Chem Phys 112 9978 (2000)





13

Chapter 2
























14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






8


τh

1 (11)

















surfaces[48] [49]












9













one time interval




( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(12)




response function

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(13)



spectroscopy[2] [13] [15] [16] [18] [19] [22] [31] [33]




10





1995)









(2002)


34 (2000)


116 2016 (2002)


110 19867 (2006)








11
















(2005)


184512 (2005)









1995)




12










Lett 86 1566 (2001)



127405 (2004)


[59] M Cho J Chem Phys 112 9978 (2000)





13

Chapter 2
























14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






9













one time interval




( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(12)




response function

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(13)



spectroscopy[2] [13] [15] [16] [18] [19] [22] [31] [33]




10





1995)









(2002)


34 (2000)


116 2016 (2002)


110 19867 (2006)








11
















(2005)


184512 (2005)









1995)




12










Lett 86 1566 (2001)



127405 (2004)


[59] M Cho J Chem Phys 112 9978 (2000)





13

Chapter 2
























14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






10





1995)









(2002)


34 (2000)


116 2016 (2002)


110 19867 (2006)








11
















(2005)


184512 (2005)









1995)




12










Lett 86 1566 (2001)



127405 (2004)


[59] M Cho J Chem Phys 112 9978 (2000)





13

Chapter 2
























14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






11
















(2005)


184512 (2005)









1995)




12










Lett 86 1566 (2001)



127405 (2004)


[59] M Cho J Chem Phys 112 9978 (2000)





13

Chapter 2
























14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






12










Lett 86 1566 (2001)



127405 (2004)


[59] M Cho J Chem Phys 112 9978 (2000)





13

Chapter 2
























14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






13

Chapter 2
























14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






14































15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






15






function







( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (21)







( ) ( )( )( ) ( )( )( )sum sum ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ sdotsdotminus

sdotminus=Π

nem mn mn

nmnmnm

mn

nmm tr

trtrtr

t 53

3 ααααα

rr

(22)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






16

( ) ( ) ( )[ ]011)3(


(23)

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(24)




[10]

( ) ( ) ( )011)3(


( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )12

12

122

12)5(

0

0

0

tt

tt

ttttR

efBPcdab

BPefcdab

efcdababcdef

minusΠΠΠminus=

minusΠΠΠminus

minusΠΠΠ=

amp

amp

ampamp

β

β

β

　　　　　　

　　　　　　　　

　

(26)


















17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






17























( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )000

0000

1

212

12212

02

)5(12

1



equiv int

BPBP

BP

t

ttt

ttttt

dtttRttW

　　　　


β

(27)

18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






18


( ) ( ) ( ) ( ) ( ) ( ) ( )00001220 2


BPttdtttR ampββ

(28)

and



( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1212

211212

00000

tttt

ttWttWttS



(210)


( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 　　　　　　　　　　

　　　　

000

000

212

112

211212



+equiv

BPBP

BPBP

ttt

ttt

ttWttWttA (211)










( ) ( )2112 ttSttS minus= (212)

and

( ) ( )2112 ttAttA = (213)







19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






19














and





112)5(

tttS tttSttR==


112)5(

tttA tttAttR==

partpartequiv of

( )12)5( ttR ( )12


)5(12

)5(12

)5( ttRttRttR AS +=












20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






20

Figure 21 (a) ( )12




21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






21















( ) ( ) ( ) ( ) ( )ωω

ωωρ 111


( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )2

2112

2112125

sinsinsin


ttCtCtCttR++

prop


int 　　　　　 (219)

22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






22


( ) 0 =ttS (220)

( ) ( ) ( )3



( ) ( ) ( ) ( )( )int

minusprop 2

2)5( cos1sin


( ) ( ) ( ) ( )intprop 2

2)5( cossin



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )int minus+propt

tRtRtRdtttR0

3335 22436 (224)

( ) ( )( ) ( )( ) ( )( )( )int ++minuspropt

S tRtRtRdtttR0

333)5( 2433 (225)


A tRtRdtttR0

3)3()5( 339 (226)



tRtRtRdt0

333 2433 and ( ) ( ) ( )( )int minust

tRtRdt0

3)3( 339 we can










expressed by

23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






23

( ) ( )tetC t 4sin4

1 222

22ζω

ζωζ minus





tRtRdt0

3)3( 339 even for















24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






24


tRtRtRdt0



tRtRtRdt0



tRtRdt0




25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






25








26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






26




















27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






27




28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






28







29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






29


























30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






30



34 (2000)


116 2016 (2002)


(2002)























31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






31



















6672 (1994)






in our simulations


32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






32

Chapter 3



IN FREQUENCY DOMAIN



















mode (NM) analysis

33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






33




























34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






34




2tqtqt abababab

λ (31)




( ) ( ) ( )tqtqtq AH λ+= (32)


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )125

125

125


where

( ) ( ) ( ) ( )12125

01 tqtMkT


and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )112

12

122125

01

001

01

tqtqtMkT

tqqtMkT

tqtqtMkT

ttR

HHHefcdab

HHHefcdab

HHHefcdabNLabcdef




amp

amp

amp

　　　　　　　

　　　　　　　 (31)


( ) ( ) ( ) ( ) ( ) ( )1212125 0101

tqtM

kTtqtM








( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2122

122125

ttCtCd

tCtCdttR

efcdab

efcdabNLabcdef



int

int

ωωρω

ωωρω

　　　　　　　

(37)

35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






35









ttR ANabcdef





( ) ( ) ( )int ⎟

⎠⎞


⎠⎞


224 2

123

2125


ωρωβ (39)







( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )int

int

int

⎟⎠⎞


⎠⎞




224

21

232

2122

122125

ttCtCd

ttCtCd

tCtCdttR

efcdab

efcdab

efcdababcdef

ωωρωβ

ωωρω

ωωρω

　　　　　　

　　　　　　 (310)


( ) ( ) ( ) ( ) ( ) ( ) ( )( )

( )⎭⎬⎫⎟⎠⎞


⎠⎞



224

21

23

21122125

ttCtCk

ttCtCtCkdttR

ANabcdef

NLabcdefabcdef

ω

ωωωρω

　　　　　　　　

　

(311)

36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






36





( ) ( ) ( ) ( ) ( ) ( )( ) ( ) int int

int intinfin infin

infin infin

=

=

0 0 125

21

0 0 125

222111125

ImIm

sinsin~

2211 ttRedtedt

ttRtdttdtR

abcdeftiti

abcdefabcdef

ωω

ωωωω

　　　　　 (312)


as

( )( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )ωωωωωωωωωω


~~~~~~~

122121120

1221211202125

ANANANAN

NLNLNLNL

RRRk

RRRkdR

+++


　 (313)

where

( ) ( )( )( )( )22

222

1

21120 42

1~γωγωω

ωωωωωω++minus

minussdot=NLR (314)

( )( )( ) ( )( )22

222

1

2

121 ~γωωγωω


=NLR (315)

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus


( ) ( )( )( )( )22

222

1

21120 42

3~γωγωω

ωωωωωω

++minusminus

sdot=ANR (317)

( ) ( )( )( )( ) ( )( )22

222

1

212

1212~


ωωω+minus+minus


and

( ) ( )( )( )( ) ( )( )22

222

1

212

122 4222

21~


ωωω+minus+minus



aωωω == 221 as

37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






37

( ) ( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+minussdotminus

+minus+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+minussdot+

+minus+minusprop

222

222

221

222

222

221

125

422

2111

422

2111~



aaaaAN

aaaaNLobs

k

kR

　　　　　　

(320)


such as

( ) ( )( )( )( ) ( )( )int +minus+minus

minusminus22

222

1

212 γωωγωω






( ) ( )( )( )( ) ( )( ) 0

222

221

21212 asymp


int intint∆+

∆minus

∆+

∆minus

ωω

ωω

ωω


ωωρωωω a

a

a

a

ddd (322)



512

512

5 2~~~ (323)



( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )aaobsaaobs

aaobsaaobs

aNL

aAN

RRRR

kk

ωωωωωωωω

ωω

2~2~2~2~

55

55

+minus

asymp (324)










38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






38




















512





As shown in Fig 32(b) ( ) ( ) ( ) ( )215



volumes ( ) ( ) ( ) ( )215





39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






39

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )int

int

minus+minus+

⎟⎠⎞


t

AN

t

NL

tRtRtRdtk

tRtRtRdtkttR

0

333

0

3335

32633

2123

23

　　　　　

(325)

and


AN

t

NL tRtRdtktdtRktR0

33

0

33 2220 (326)








lines

40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






40


512











41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






41




( ) BrAr

rU +⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

46

σσε (327)












5 ~ ωωR and ( ) ( ) ( ) ( )215




42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






42







respectively[36]


( ) ( )125 ~ ωωR and (c) ( ) ( ) ( ) ( )21

512





43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






43


( ) ( )125 ~ ωωR and (c) ( )( ) ( )( )21

512













44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






44









45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






45























modes respectively




46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






46





( ) ( ) ( )tqgtqqV 3322

32+=

microω (328)








( )1

21

12sin11

ln1C

CtDCCtq

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

=

βmicro

β (329)

and

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+

⎟⎟⎠

⎞⎜⎜⎝

⎛+minus

=

21

1

21

1

12sin11

2cos12

CtDCC

CtDCCDCtp

βmicro

βmicro

micro (330)




( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus⎟⎟

⎠

⎞⎜⎜⎝

⎛+minus= L2

2121

2sin22


βmicroβ

(331)

and

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minusminus⎟⎟⎠

⎞⎜⎜⎝

⎛+minus= L2

121

22sin2


βmicro

micro (332)


47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






47

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+minus+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

partpart

LtDCtDCtDDp

tq βmicro

βmicro

βmicromicroβ 2

sin2

sin142sin21

03

21 (333)

Thus

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= 2

1 2sin1

CtDCtqH β

microβ (334)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++minus

minusminus= 2

21 2sin22

1 CtDCtqA βmicroβ

(335)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= tD

DtM H β

micromicroβ2sin

21 (336)

and

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

minus= tDCtD

DC

tM A βmicro

βmicromicroβ 2

sin2

sin2

14 32

1 (337)



( ) ( ) ( ) ( )

( )

( )　　　　　　　

　　　　　　

　

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝


21231

1221

21221

125

22sin

2sin141

2sin2sin11

2sin2sin11

tDtDtDCkT

tDtDCkT

ttDtDCkT

ttR

efcdab

efcdab

efcdababcdef

βmicro

βmicro

βmicromicroβ

βmicro

βmicromicroβ

βmicro

βmicromicroβ

(338)


becomes

( ) ( ) ( ) ( )

( ) ( )

( ) 　　　　　　　

　　　　　　

　

⎟⎠⎞




2123

2

122

2122125

2sinsin4

sinsin1

sinsin1

ttt

tt

tttttR

efcdab

efcdab

efcdababcdef

ωωωωβ

ωωω

ωωωω

(339)



and Stratt

48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






48


1995)
























(2005)




49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






49

334 (2000)


116 2016 (2002)






(2002)





A 263 378 (1999)


50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






50

Chapter 4



CU(100)


















51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






51
























response function

( ) ( ) ( )[ ] ( )[ ] 0 12

2


⎠⎞


(41)




52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






52

( ) ( ) ( )[ ] ( )[ ]12

2


⎠⎞

⎜⎝⎛=h

(42)



















53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






53



( ) ( ) ( ) ( )1212)5( 01 tt







( ) ( ) ( ) ( )

( )( )

( )( )

( )( ) ( )1

2

2

2

1212)2(

00

01

01

tqp

tqtqt

kT

ttkT

ttR

cij j

b

j

i

i

a

cBPbaabc

minuspartpart

partpart

partpart

minus=

minusminus=

sum micromicromicro

micromicromicro

amp

amp

　　　　　

(44)

















54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






54






























than the NEMD

55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






55







42










56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






56









tone area ( )12

111


( )12

111


respectively[30]






2 21792123 minus= cmω



57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






57














58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






58































59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






59



interaction





60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






60

( ) ( )( )

( ) ( )( ) 21222

222

2

2122211

22211

2

cossin41

coscossinsin41

θθπε

θθθθπε

rrR

q

rrrrR

qV

++minus

minus++minus=

　　　

(47)














61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






61












62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






62














respectively














63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






63



























can be rewritten as

64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






64


where

x

xiLK partpart

= amp (48)

( ) p

xFiL sVs partpart

= (49)

and

( ) p

xFiL lVl partpart

= (410)


where nTt ∆=∆








322



TiLntLLLiTiLTiL

VlKlKsVsKsVl

VlVsKsKlVl

∆+∆+=

∆+=∆∆∆∆∆∆

∆∆++∆∆

(413)









65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






65







65 480 (1990)



214 431 (1993)


101 1704 (1994)







(2002)






New York 1995)



1984)


66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






66


[24] M Cho Phys Rev A 61 023406 (2000)


Lett 86 1566 (2001)




127405 (2004)

[29] M Cho J Chem Phys 112 9978 (2000)




















Chem Phys 120 10215 (2004)


67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






67





(1990)


(1991)









1338 (2005)


Chem Phys 125 244508 (2006)




Pro Nat Aca Sci 102 13019 (2005)

68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






68

Chapter 5

CONCLUSION
























69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






69












through Eq (324)















70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






70




















formalism











71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






71

linear spectroscopy



34 (2000)


116 2016 (2002)


110 19867 (2006)


(2002)







110 19867 (2006)






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